Depending on which route you choose, you start out as either Alfred Arrowsmith, after his united front with Rick against Gustave, or Kuki Youkou after his final confrontation with Soushichi. Ungaikyou (literary means the mirror beyond the clouds), a youkai made from the failure attempt of the mirror of the three artifacts belonging to Amaterasu (the Sun god in Japanese mythology), has chosen these two to help them create a world of their own. In exchange, Alfred and Kuki are temporary exempt from going to the netherworld and finish their lingering desires…in their non-respective worlds.
After the completion of the two stories, the final route—Chronobelt—unlocks. Almost immediately after Kuki and Alfred finish dealing with their lingering desires, invaders have slipped through the portals and into the Infinite Corridors (the space connecting the two worlds). Before long, team Ayakashibito is in a constant war with team Bullet Butlers in what can only be described as the city of Chaos.
[Edited from a really spoiler-filled review by Warum]