In the following account, I present my proposition for two new neutral opposing factions:
What I am about to elaborate on here first is the connection to Tolkien's lore and the reason why I believe the two factions that I propose fits the lore. The first part of the connection involves Dirhaval, the character who wrote the in universe "Tale Of The Children Of Hurin" and "The Wanderings Of Hurin". We do not know anything about Dirhaval besides his writing and that he was killed during the third kinslaying. We only know that he wrote his writings because he felt that the history of Hurin and his children was nearest to his heart. While we know that this is why he was interested in the Narn, it does not explain why it was near to his heart. We also do not know the specific impact and influences that the Narn had in later ages, only that it was the greatest and likely was highly influential. The second part of the connection to the lore involves the fate of Hurin and the fact it is technically unknown what happened to him in the First Age. It is only said in "The Silmarillion" that he MIGHT have committed suicide, but it is not outright said. Though it is highly likely that he did so. Therefore, my theory takes that into account along with Hurin's mental state within "The Wanderings Of Hurin".
My interpretation is that Hurin discovered that due to his mental torture at the hands of Morgoth, and the revelation from Melian that everything he does benefits evil in some way, drove Hurin into a despairing madness and as a result, he commited suicide because he could not live with the knowledge that every action he did served no purpose beside benefiting evil. Now that is the lore that I was influenced by. Now, shifting attention back onto the in universe "Tale Of The Children Of Hurin," it is my theory that due to Dirhaval's work being said to be the greatest of all the tales of Beleriand, that future loremasters and writers (ie. Elrond, Bilbo, Findegil, etc.) would discuss, comment, reveiw and critique Dirhaval's work. The two factions that I am about to propose are not factions in the typical sense and more of two large "cults/organizations/guilds".
The background of these groups is that two philosophical traditions emerged from readers of the Narn: Hurinism and Umarthism. The thinkers of Hurinism are known as Hurinists/Hurinlings and the thinkers of Umarthism are known as Umarthists/Umarthans (ill-fated). The Hurinlings began with Dirhaval, believing that the tragic story of Hurin and his family is about how even the greatest heroes are not infallible and that even those who have fallen so far can still achieve great things (The Death of Glaurung) despite the pain they torture (The Wanderings Of Hurin) themselves with. The Hurinists believe that nobody (however far they have fallen) are beyond redemption. The largest dwelling of Hurinlings is mostly on Tol Morwen, Tol Fuin, Tol Himling, but they can be found anywhere as they have no ties to any land (though they once tried to maintain and preserve the Stone of the Hapless on Tol Morwen where Turin and Morwen are buried). Their largest settlement is called "New Dor-Lomin" located on the south eastern half of Tol Fuin.
The Umarthans of Umarthism began much later and they derive much of their philosophy from Hurin's final days when Melian revealed to him that everything he did was pointless only to benefit evil. They believe that free will is an illusion and that every living thing is ill-fated in that life has no meaning so way bother resisting evil? They reinforce their belief with the speculative idea of the Dagor Dagorath and Arda Unmarred. With this, they believe that their only purpose would be in preparing Arda for the Dagor Dagorath in order for Turin to avenge Hurin and for life to become meaningful in Arda Unmarred. The Umarthans and Hurinlings both despise and look down on each other, with the Hurinlings trying to redeem the Umarthans and the Umarthans trying to eradicate the Hurinlings. This resulted in the Hurinlings being driven from all three islands with only "New Dor-Lomin" still within their control. The Hurinlings eventually began to call the land of the Umarthans the "Nan Dumgorthin", which translates to the "Land of the Dark Idols" due to the Umarthans habit of building elaborate idols of scenes from Dirhaval's Narn that reflect their philosophy, as well as idols to the Dagor Dagorath and the speculative confrontation between Turin and Morgoth.
This is only a rough synopsis of my proposition. If you have any questions or see an issue with my proposition, please feel free to comment and let me know.