This has most likely been done before, but here's an idea for an AU.

Yuri, Rin and Yukio's mother, passed away at birth, which was of course devastating for everyone. They were supposed to live a happy life together as a family in Assiah. And prove that demons and humans could live together in peace. But thanks to the exorcists, that was no longer possible. After Yuri's death, Satan tried to take both babies back with him to Gehenna.

Except Fujimoto and Mephisto showed up to stop him. Satan was only able to leave with Rin before the portal closed. Fujimoto got hold of Yukio. 

Satan tried countless times to get Yukio, but was stopped by Fujimoto every time.

Yukio ended up being raised in Assiah by Father Fujimoto and Rin was raised by Satan in Gehenna.

Later on Yukio is told about his twin brother. And how he was taken by Satan, their real father, at birth.
Yukio hopes to someday be able to rescue his twin brother.

Rin is told a similar story by Satan growing up. How he had a twin brother, but he was taken by the evil exorcists at birth and they had to try their best everday to get him back.

Since being told of his brother, Yukio spends just about every day worrying about the kind of torture he could be gone through living in Gehenna. Or if he was raised to be evil, being that he was raised by demons.

Despite what Yukio thinks, Rin grows up to be perfectly fine. And lived an alright life in Gehenna.

Eventually, Rin is allowed to go to true cross which is run by his older brother Samael (Mephisto). He joins the exorcist class to finally meet Yukio and they are reunited.