Hagoromo took back his Yin and Yang chakra from Sasuke and Naruto respectively (in chapter 690). So if the Yin chakra was responsible for the Rinnegan development in Sasuke's eye, why didn't it change back into his original eye after the chakra was taken back? Why does it become permanent?
Well, we have seen several instances where even losing Six Paths Power still makes one stronger than most, as described by Kakashi after he obtained Obito's power. I don't remember the exact chapter, but right after he awakened Sharingan in his natural eyes from Obito transferring his power, Kakashi says something along the lines of the eyes being stronger after Obito played host to the Ten Tails. One could also argue that having the completed chakra of the Six Paths was only a requirement to awaken the Rinnegan, not keeping it. However, I must stick with whatever facts that I can see and those are thus; but first.
Spoiler warning, just in case you haven't read past chapter 690 as mentioned in the OP's question.
In chapter 696, after Kaguya was defeated and the Six Paths Yin and Yang symbols had returned to Hagoromo, we witness the final fight between Naruto and Sasuke at the Valley of the End. In this fight, Sasuke was still wielding his Rinnegan, but more importantly for the evidence involved, Naruto was still using Six Paths Sage Mode. That being said, it leaves little room for interpretation that the two of them must still have Six Paths Chakra of their own.
Now the room we do have for speculation comes here where we must decide whether they still have these powers, because
A. When Hagoromo regained the seals on his hands, they were just the seals for the Six Paths Chibaku Tensei and nothing more. Or
B. Even though Hagoromo took his powers back, having been exposed to the Six Paths chakra has awakened their natural ability to use it, since Sasuke and Naruto were the reincarnated Indra and Asura respectively. After all, Madara was able to awaken the Rinnegan by introducing Hashirama's cells into his own body unknowingly, which means introducing Asura's chakra into himself. After it was integrated, he didn't have to continuously inject Hashi's cells to keep the Rinnegan.
Really, Sasuke ' s Rinnegan was awakened the same way Madara's Rinnegan had been awakened. When Sasuke wakes up from his talk with Hagoromo, we see Kabuto healing him using some of his own body's regenerative properties.
Kabuto had, long before this, already fused Hashirama ' s cells in his own body and by fusing his flesh into Sasuke, that once more uses Hashirama to combine Indra and Ashura's flesh which is what awakens it.
However as Madara said, it still took years for the flesh to take effect, so likely the exposure to Hagoromo's Chakra in his seal he gave them managed to jump start the effects.
I don't know for sure, but I would think it would make sense that having such a long process suddenly happen in just those couple of minutes could have complications or mutations, which may be why Sasuke only got it in 1 eye, and why it's in the same side his yinyang seal is held.
Given that, it's possible the right side may eventually catch up and activate. Also, after Madara activated his own Rinnegan, he's never seen deactivating it back to his normal Rinnegan until he retrieves it from Obito and Zetsu, whom got the eyes from Nagato. So it may not be something one can so easily deactivate like the normal Sharingan eyes. He may have to awaken and master all 6 paths before he can deactivate them.
Considering these facts and any other complications that could have arisen from the Sage's abrupt interference with the natural development, and Sasuke may never be able to properly activate and deactivate a matching pair of eyes whether they're Sharingan or Rinnegan. (Except perhaps when he's much older, like grandpa to Sarada kids older)
the reason sasuke still has the rinnegan is actually really simple. once you have it, its permanent. look at madara for example, he unlocked the rinnegan before he died, and before he had hashiramas cells.yet after he became an edo tensei, he was still able to use it after re obtaining his eyes from zetsu.
Hagoromo's Yin power unintentionally became a catalyst to speed up the Rinnegan forming process, as stated by Madara when he awakened the Rinnegan prior to his death, meaning that it takes a long time for the Rinnegan to form. Madara even told Obito "[...] even if you don't gain the Rinnegan [...]", and this was after he healed Obito with White Zetsu's skin. The presence of Hagoromo's chakra shortened the Rinnegan formation process for Sasuke, and even then it was only in his left eye. I believe his right eye will become a Rinnegan later in his life.
I agree with most of the reasons people put. But i think the reason why Sasuke kept his rinnegan was that when he and Naruto last fought, they lost their dominant hand but then started bleeding. As they were bleeding, their blood join and kind of mixed part of Naruto's and Sasuke's chakra together. I don't know if this is correct or not so ima just get it from memory. I remember that somewhere in the Naruto Shippuden series, someone said that when a Uchiha (Indra's descendants) obtains the chakra of a descendant of Ashura. The user unlocks the rinnegan. Im pretty sure that Madara didnt get it right away because he only had a little peice of Hashi's chakra.
i would like to talk about the truth. Sasuke's Rinnegan has nothing to do with hashirama's cells
as hashirama ceased being a transmigrant upon his death. hagoromo said that hashirama and madara were the previous transmigrant. key word WERE they no longer are transmigrants Naruto and sasuke are the current transmigrants as such Hashirama's cells no longer carried the potential to awaken a Rinnegan
sasuke's Rinnegan is solely because Hagoromo gave Sasuke six paths power. naruto's six paths senjutsu is solely because hagoromo gave his six paths power to naruto. now people assume he only gave the each half of his power but this is a lie he gave them both both halfs of his power. the only reason Naruto didn't gain a Rinnegan was because he doesn't have a sharingan. and like wise sasuke only has 1 Rinnegan because he lacks the senjutsu of the six paths if Naruto and Sasuke were to exchange DNA Sasuke would awaken a second Rinnegan, Naruto's body would than acclimate to the full required genetic halves of the sage of six paths and eventually awaken a Rinnegan