Why do Indra and Ashura reincarnate?

Question

The anime tells us that Madara-Hashirama and Naruto-Sasuke are the reincarnations of Indra and Ashura.

Why did they get reincarnated? For fulfilling what purpose?

Answer 1

There is a lot of debate that the incarnation cycle was started because Hogorama knew that Kaguya would one day finally be free of their seal and restart her god-tree ritual which would consume human-kind. But that is just in terms of giving the plot a direction.

Speaking scientifically, energy can neither be created nor destroyed. It can only be transformed. Kaguya initially attained massive amounts of energy due to being a member of Otsutsuki and the energy grew exponentially larger after she consumed the chakra-fruit. When she had kids (Hogoromo and Hamura) they inherited a part of her energy. After sealing their mother, the only humans who wielded destructive energies were Hogoromo and Hamura. But Hamura left earth to live on the moon to keep guard on their mother and the elder brother Hogoromo stayed on earth and had two kids Indra and Ashura who inherited a part of his energy. They later grew up to become able-bodied adults who could use Hogoromo's power or Nin-Shu (not Nin-Jutsu) as he called it. While Indra used that power in a destructive way, Ashura, on the other hand, used it to help people forming the earliest concept of Yin and Yang. They were beings of enormous powers and when they died their energies or power were transformed into their incarnates and like opposite poles attract, so were the incarnates drawn towards each other.

It is also evident that Hogoromo could transcend his consciousness over time. It was just that his energy was not being transformed and remained as a silhouette over generations.

Answer 2

There isn't really a well defined answer, but there are a few key plot points that provide hints.

First and foremost, Hagoromo himself. He is dead, and yet can still manifest in the living world and even give chakra out. He is half Ootsutsuki who have many mysterious powers, so its not all that crazy.

Second, Indra and Ashura are Hagoromo's kids, so them inheriting mysterious powers is a given. If Hagoromo can do it, they probably have some ability to do so as well. Hagoromo though has the Rinnegan which controls life and death, so him having more or less complete control compared to them would also make sense.

Third, Obito, who had a dimensional shifting power and could use chakra, which in his own words "connects the two worlds" to return back and even give chakra to someone. If that sounds familiar, its because its exactly what Hagoromo did. He however had dimensional shifting powers, which explains how he would be better than Ashura and Indra.

Fourth, the Rinnegan being able to revive people, and the Edo tensei being able to pseudo revive someone, as well as Chiyo's sacrifice to revive Gaara shows Obito's point about chakra connecting the world of living and world of dead is correct.

So, Ashura and Indra have not only some of the mysterious powers of Hagoromo, but they also have an incredible hatred of each other due to what happened thanks to Hagoromo. They fought, probably to the death that day. Indra believed power, and was defeated by his younger, and he believed weaker, brother. its a great insult to his pride. Power is wrong, he is wrong, and the weaker one beat him not only by inheriting Hagoromo's legacy, but then in battle. He probably wanted to win in the long run, so he transmigrated so Ashura would be long dead and unable to stop him, so Ashura did the same to stop him. A blood feud that even death can't stop.

Answer 3

It's just that the seal that they applied on their mother was not fully perfect, as they knew a day will come when their mother Kaguya will revive, so they had to revive as well.