I thought it was more that the power of Shadar Logoth cancelled out whatever the Shadow imposed on him, the way Rand cut Asmodean off from the Dark One. It doesn't really fit for him to be the third axis if he is still part of the Shadow, or for the two types of power to coexist in the same entity without destroying one another and their host. I'd think then that the power of Mordeth & Shadar Logoth replaced whatever Shadow connection existed in Fain.

Unless I am misunderstanding this, and you are contending that Mashadar and Shadar Logoth are also Shadow stuff. Actually, that might make a degree of sense, given what Aginor said about Mat's dagger, when he revealed that he and Baalthamel tracked the Eye by detecting the dagger, and referring to it as if its power was known to them already. I don't recall what exactly RJ said about the difference between their imprisonment and that of the others. Did he say that they were aware of the world and passing events? If he did, that could mean what Aginor meant, that he had observed the fall of Aridhol, and was recalling it. Actually that might make more sense than the Shadar Logoth power dating back before the Breaking, since to a Forsaken who just woke up, it would not have been all that long ago.

Or become one.

Maybe entropy, too? That might make sense as both the opposing force of the Shadow that makes them cancel one another out, AND as something that originated with the Shadow in the first place. Maybe SLpower is some inversion or twisting of the ShadowPower into its opposite. Anti-Shadow as in anti-matter, the complementary, or supplementary aspect of the Shadow.

I wonder if the philosophical mechanism has to do with the Shadow's habit of self-destruction or self-consumption, that the Dark One concentrated on "positive" ShadowPower of perversion, corruption and destruction, and separated it from the entropic and corrosive aspects of the Shadow, so as to use it actively.

There does not appear to be any overt manifestation of the Light and the Creator unless maybe Rand's post-apotheosis shielding from the Taint. Thus, without interference, the Light and Shadow are relatively inactive powers or forces, the Light simply being in perfect balance and harmony, while the Shadow rages against and consumes itself, and thus gets in its own way. Then the Dark One, maybe takes hold of the more useful (in his eyes) and overt aspect of the ShadowPower (I am using that term to differentiate between the True Power as wielded by human beings) to offer to corruptible humans, and exploit the breach into his prison. He uses that just about exclusively to exert his will on the world, because to use the other Dark power would simply undo or interfere with his own works. From his preferred power, he gives his Chosen the True Power to wield and make the Shadowspawn and whatnot, and that is what he uses for his counterstroke against the Sealing by LTT & co., so that is how saidin is tainted.

It could be too, depending on if you believe the One Power is to the Light as the True Power is to the Shadow (I prefer to think of it as an unaligned neutral resource, no more aligned with the Light or Shadow than are fire or gravity, otherwise, there should be some sort of issue with the Forsaken, at the least, using it), that ShadowPower and SLpower are the Dark counterparts to saidin and saidar, except rather than ideally working in tandem and complementing one another, they would cancel and counteract one another, because the Shadow is inherently inimical to everything, even itself.

Given the pseudo-scientific functionality of the powers, I would think it would more likely involve a mentality or outlook, a kind of putting oneself in alignment with the source of the Power. It wouldn't be a ritual as such, with an invocation of another entity, but there might by symbols or phrases or such things to help put one's mind in a frame of thought so as to be compatible with such a power.

Or maybe, he was trying to create a critical mass of attitudes and feelings amenable to his type of power. Just as belief and order in some fashion lent strength to the Seals on the Dark One's prison (and the sorts of degeneracy in places ruled by the shadow - it can't be mere license that manifests such extremes, it seems as if active evil is encouraged as opposed to permitted. Perhaps the Dark One or the Shadow feeds off of wrongdoing and gains strength from corruption and destruction), maybe the paranoia and perversion of ideals did the same for Mordeth with his alternate power.
Actually, I think that makes a lot of sense. If the Dark One is imprisoned outside the Pattern, and the Pattern is formed of lives, and thus the actions and deeds of people, it makes sense that if a significant portion of the Pattern is directly aligned to the same values and perspectives and outlook as the Shadow, the Dark One could enter the Pattern, or join with it. Atrocity, corruption, defilement and destruction all contribute to an atmosphere or environment into which the Dark One is no longer an intruder, but a natural denizen. Thus Mordeth was doing the same thing in Aridhol as the servants of the Dark One, especially Elan Morin, would have been doing in the AoL during the Collapse, only he was encouraging his own perspective, that fits better with his own power.

This too would explain the attitudes of the cities in which a Forsaken dwells or reigns. They are connected to the Dark One and are already steeped in his evil, so they are vectors who accelerate that saturation of evil into the Pattern. Rahvin, the insecure, caused Caemlyners to start scheming and become furtive. Sammael, the frustrated, caused tempers to flare, and people give rein to violence. Bel'al, the envious, spread despair, which seem to be a logical result of universal envy of each other, the ultimate end of dissatisfaction. Moghedian's personal avarice for material gain and physical cowardice might have contributed in some way to the lawlessness of Tanchico, but it could also be that the chaos concealed the obvious single symptom. It could also be that without her exerting any influence over the rule of the city, she was not having the usual Forsaken effect. And my personal theory, of course, is that Mesaana was causing the tolerance of filth in Tar Valon. Her characteristic was thwarted ambition and inadequacy, so it could translate to people starting to dwell more on their dreams and yearnings, and losing interest in basic daily maintenance of their present environment.

Anyway, back to Mordeth doing something similar to work Aridhol into a mass attitude of anti-Shadow mentality. Because he was the one wielding it, instead of unconsciously contributing to its generation, it would explain how he retained his consciousness and will, while the other minds would have been absorbed into a mindless appetite, which seeks to consume as the Shadow seeks to destroy.

I think that supports the point that they were always there, they were simply inaccessible. It seems kind of a game breaker for one side to be able to impose a change that can't be undone by what exists in creation. But because the Dark One only struck with his side of the Dark Power, he left his work vulnerable to the other aspect, since their purpose was to consume/destroy one another. The Dark One would have HAD to leave it out to have the effect he sought, since as we see in Rand's wounds, the two types of evil are mutually limiting. Had both aspects, SLpower and ShadowPower been used against saidin, it might have been a bad experience for the men who touched the taint, it might have given them tendencies to evil and darkness, and some physical side effects, but not the directly harmful madness and destruction of the body that actually occurred.
I long had the theory that the Black Wind was Mashadar/Shadar Logoth influence on the Ways through the Waygate in the city, until I saw some comment by RJ that seemed to say otherwise. And now that this has me thinking, overtly bloodthirsty madness is more of the Shadow characteristic. A Shadar Logoth wind would be encouraging hubris or greed on people as well as paranoia, not destruction or cruelty for its own sake. Yes, Fain was mad, but that seemed more of an effect of cumulative trauma and exposure to horrors, than mere supernatural influence. The "professional courtesy" between Fain & the Black Wind might then be related to their mutual heritage of the shadow.

Or maybe it was the contradictory aspects of the evil power inevitably drawing together, and regardless of which of them got to Rand, they would then spend an eternity grappling with one another, Shaisam trying to consume a Dark One that is too much for him, and Shaitan trying to destroy a hostile force that is too much like him to completely do away with, and which is feeding from him and recovering its strength at his own cost. Note as well the similarity of their names, like saidar and saidin: two complementary aspects of a greater whole. If Rand fell to the Dark One, Shaisam would still have consumed him as a prelude to confronting the Dark One, or the Dark One would have used him to defeat the avatar of Shaisam, to retain his monopoly of sentience within the Dark.

I think it would more likely have simply deprived the Dark One of a powerful tool against the Pattern. As I said above, the more the world is made akin to the Dark One's ethos, the more access he has, and more influence. If the mere Forsaken can have such an effect on a city in so short a time, an equivalent individual, who is also a very powerful ta'veren can do so much more to spread evil into the Pattern. This would explain HOW Ishamael sees Rand as the key to the Dark One' ultimate victory, and incidentally his own release: with Rand doing his bidding, the Dark One would discard his former best servant. RJ did say the Dark One's retention and favor of the Forsaken was strictly utilitarian. It was not for the troops they could command, or deeds with the One Power they could perform, or the enemies they could slay, it was how they spread his influence into the Pattern. A guy whose mere existence can re-order the world, would render the other Chosen pointlessly redundant.

I think the choice to destroy Shai'tan would have made Rand vulnerable to such an attack. It would be an act of hatred, of negation, rather than an act of love and appreciation, and one way or another, negative feelings is how the Evil gets you (especially hostility toward the Dark One, which is the sentiment that brought down Aridhol). Even if the entity known as Shai'tan would have been vulnerable to Rand in that moment, Rand in turn would have been vulnerable to Shaisam, who might then have Rand as a subject/plaything he could use to spread HIS version of evil through the world and the Pattern. That would not preclude his ongoing torment and consumption of Rand through the process, and Rand's hatred and resentment would probably only enhance the effect.

For reasons mentioned above, I think it's more like they are both similar creatures borne of different aspects of the Evil that includes Shadow and Shadar Logoth. IIRC RJ has directly asserted that the tainted saidin used to grow the Ways was the source of Machin Shin. Only instead of subsuming its victims as Mashadar does, it adds them to the cacophony and breaks them down and drives them mad.

I doubt annihilation was what the Dark One had in mind. He would have been remaking the Pattern in his own image. Shaisam might take his place, and it would be a different brand of Evil, but effectively the same.

For my own conclusion, I think the Dark One preferred one aspect of the Evil power, selecting the pro-active one for its obvious utility, but by their nature, had to use it exclusively. Or maybe he was naturally aligned with that aspect, and one of the differences between the two is that the Consuming Power had less need for, or compatibility with, a sentient mind, being at the base, an appetite. On the other hand, the Destroying Power, would need a mind to guide it in destruction, to provide the motivating hatred. Either one would make a sufficient explanation: the supernatural Evil, or Dark, which is the counterpoint of the Light, has two complementary or contradictory aspects, one of which is a power to destroy and the other a power for consumption. While the benevolent Light rests apart from the Pattern and material world in perfect balance, the Dark which would consume or destroy it, is held in check by its own contradictory natures. Just as the Light proceeds from, or manifests the persona of, a benevolent Creator, the Dark also manifests a persona, who is attuned more closely with the Destroying Power of the Dark, whether by nature of the DP, or by his own preference as the most practicable to effect his hatred of the Light's creation. Because the Light IS harmony, it is impossible for there to be harmony in the Dark, and thus Shaitan, the persona of the Destroying Power, cannot master or balance both the Destroying and the Consuming Power. So the Dark One eventually finds his way to access the Pattern and uses the DP to corrupt and degrade the Pattern and remake portions of it in his own image, while the lives that make up the Pattern flail about in ignorance, trying to contain that damage while being unable to comprehend the scope or nature of what is doing this. The Light does not intervene, not because it is constrained, but by its very nature and the nature of the Dark. Destroying destruction itself would be a paradox, and might very well unleash an unstoppable tide of the Consuming Power without the Destroying Power to thwart it. But the Pattern is a self-correcting feedback mechanism. To the extent that the Light counters the Dark, it is with the built-in feature that allows its creation to resist and work against a threat to the system. And so, with the DP (or "Shadow" as the Power and its user, Shai'tan are collectively known) running rampant, the Pattern opens up a way to introduce the DP's antithesis into the system, and allows a sufficiently vile individual of the appropriate personal characteristics to be compatible in a fashion with the Consuming Power, to come to exist. That is, of course, how it works with ta'veren, by using an individual with the necessary personal qualities to affect many others. Likewise, it threw up the necessary combination of factors to allow the CP to manifest directly in the world as the Forsaken and their compatriots brought the DP. The initial intrusion is sufficient to tip the balance against the Shadow during the period of the Trolloc Wars, but then with its counterpart force receding, and humanity much depleted, in both numbers and moral strength, the CP lacks anything to consume and spread, and lies inert, waiting for the chance to consume more. The odd traveler and adventurer are mere crumbs, and it devours them before it can make use of them to move beyond the place where it came through the Pattern, because that atmosphere of paranoia and antipathy is so rare. There are many ways for destruction to manifest, while consumption by its nature is inherently limited. The avatar of the CP is trapped and bound in a semi-corporeal form to a limited location, just as the partially sealed Ba'alzamon is, and Shaidar Haran will later be. This avatar, Mordeth, seeks to move beyond its physical limits, by joining with a physical form, made compatible by appealing to appetites, at the limits of its ability to manifest. It finds a mortal marked and claimed by the DP, and thus is unable to thoroughly consume him, forcing a merger, but allowing the gestalt entity to resist the Dark One's power in the same way a Shadow-wrought wound and Shadar Logoth inflicted wound cancel each other out to a degree. The dagger, which carried a portion of the taint out of the locale to which it was bound, tips the balance away from the DP in Fain, and allows him to begin asserting himself over Darkfriends and Shadowspawn. His powers grow so that losing the dagger doesn't cripple him, but its recovery still brings him closer to his own apotheosis as Shaisam, avatar of the CP. His hatred of Rand, Mat and Perrin is how his recognition of their utility to spreading HIS version of the Dark manifests, and he seeks to hurt them to satisfy his own broken nature and to break them and harden them so they can turn more people into paranoid, hateful, anything-is-acceptable-against-a-foe, and draw more of the Consuming Power into the Pattern. Note that nearly everything he does is focused on stirring enemies of Rand, and ginning up hostility towards him. He likely had his own version of 13x13 in mind for the ambush in Far Madding, where Rand could not use the Power to harm him. I wonder if maybe Tainted saidin might have been even more effective on Fain. And then Rand cleanses the Taint of the DP on saidin and the taint of the CP on the ruins of Aridhol, by forcing them into contact with each other. As fragment separated from the whole, they were vulnerable to mutual destruction since only a defined & finite quantity of each was involved. Both Powers would necessarily center around the individual who sentience guided them, with the Fain/Mordeth gestalt as the new wellspring of the CP. Shaitan is too powerful, too united with the DP for him to have the same mobility in the world as Ordeith/Fain/Mordeth.

Thus Shaisam can serve as the wild card in the confrontation between Rand and Shaitan. If Rand wins with evil, it brings him into its fold. If Rand is losing to the Dark One, Shaisam might tip the balance enough for him to take advantage and survive. If the Dark One is somehow eliminated, Shaisam is there to take his place, as the new Great Lord of the Dark oriented around the Consuming aspect of the Dark Power.

Too much work. Draghkar suck the life force out too. The Shadow has other ways of doing that sort of thing, like the 13x13 or the mindtrap or that game they mention with playing pieces that were once human, or the Myrdraal sword forging process.