Why does Sasuke still have the Rinnegan? 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.