We know that the Rinnegan is the next stage of Sharingan and since Madara can switch from Sharingan to Rinnegan and vice versa, why can't Nagato?
Kakashi received the Sharingan at a young age. He had years of experience with it and was considered a genius at the usage of Sharingan. He was even appointed to train Itachi and Sasuke Uchiha with their Sharingan development. Even with the amount of knowledge and power of the Sharingan, he couldn't revert the Sharingan back to its normal state. That was because of a simple reason: the eye wasn't his. It originally belonged to Obito Uchiha. Since Kakashi wasn't the original owner of the eye, he is unable to revert the eye back to normal.
Because he is not an Uchiha, Kakashi was unable to deactivate this dōjutsu.
The article tells us that, in order to deactivate the Sharingan, the person needs to be an Uchiha. This can also be related with Danzo Shimura and his arsenal of eyes.
Obito "borrowed" the Rinnegan from Nagato, who originally got them from Madara Uchiha. There were many reasons why Obito implanted only one Rinnegan eye even after having obtained both. However, one of the main reasons was:
Under the guise of Madara and inheriting the former's work, Obito Uchiha claimed that he was entitled to take the Rinnegan from Nagato's body after his death. He then implanted the left Rinnegan into himself and hid the right Rinnegan, unable to handle the power of both eyes.
Since Obito wasn't the original owner of the eyes,
- He did not have complete mastery over it.
- He couldn't handle its overwhelming power.
Obito, being an Uchiha, lacked control over the Rinnegan. This kind of proves that an Uchiha cannot simply control the Rinnegan if he isn't the original owner. Keep in mind that Obito was trained by Madara himself.
Nagato received the eyes at a very young age. He did not know that the Rinnegan were transplanted into him, and nor did he know that it evolved from a pair of Sharingan eyes. Nevertheless, he had almost complete mastery over them because:
- He was an Uzumaki and thus had huge chakra reserves.
- The eyes were implanted at a very young age, giving him time to perfect his Rinnegan skills by adulthood.
Sakurai Tomoki's answer is logical and makes sense. However, if we were to look at similarity between the Rinnegan (evolved from Sharingan) and the Sharingan itself, it seems that answer could be different.
We know from Kakashi's case that he couldn't deactivate the Sharingan because of its origins. We also know that Obito lacked control over the Rinnegan, which implies that he couldn't deactivate and revert it into a Sharingan. Combining these two factors, we can derive an explanation to the original question: Nagato couldn't revert the Rinnegan to Sharingan because he wasn't the eyes' original owner.
Madara displayed certain new and powerful techniques when he received his original eyes.
Because Madara was the eyes' original owner, only he could use them to their fullest power.
In many situations when the power of a borrowed dojutsu is questioned, the full potential of the eyes is always attained by the original owner. So it's logical to conclude that the Rinnegan to Sharingan swap was exclusive to Madara and not to the other two.
The reason are most likely similar to Kakashi's Mangekyou Sharingan case. He just simply doesn't realize it. Nagato "awakens" his Rinnegan when his parents are murdered by the mistaken Konoha shinobi. But actually it was Uchiha Madara implanting it to him, not him awakening the Rinnegan. So, while he can use Rinnegan to great extend, he simply doesn't know that it was originated from the Sharingan and that he can revert it back to Sharingan.
This is the same as Kakashi. Although he has awakened his Mangekyou when he "killed" Rin, he simply doesn't realize that until years even though he was dubbed as a genius ninja.
Also it needs to be noted that the ones who knows about the Sharingan can be upgraded into Rinnegan are those who has awakened the Eternal Mangekyou Sharingan and read the stone tablet in the Uchiha underground hideout using Sharingan, Mangekyou Sharingan and Eternal Mangekyou Sharingan in order. Nagato, being a non-Uchiha and lived far away from Konoha, would have no knowledge of this.