Spoiler (mouse over to view)
The main story that we're reading with the english-translated scans is a flashback of their past life up until chapter 18. Basically, Karuta was cursed and she couldn't go back to human form. The bad guy (whose name directly translates to: "Dog God") is trying to create a revival/ceremony that involves the half-demons. Soushi, Kagerou and Nobara were involved in the fight and at the end, Soushi died while protecting Ririchiyo. And now we fast-forward 23 years, where all the characters are reborn in their second life, with the exception of Renshou, the sole survival of the last battle.
In the new timeline, it begins where everyone except Ririchiyo, Soushi and Watanuki has reclaimed their memories of the previous life. Soushi and Ririchiyo meets again, and Soushi seemed to be acting like the original Soushi - except that later we find out that Soushi only felt indebted to Ririchiyo from saving him from his household. He went to research about the behaviour and personality of the old Soushi, and chose to impersonate his past self as he felt that his old self was the most important person to Ririchiyo. Meanwhile, Ririchiyo slowly reclaims her memory by meeting Renshou and the others (ah, at this point, I'd have to admit that it's very touching - Renshou was a brother-figure in her last life, but now he's almost like a father-figure since he's 41 years old). Anyway, they decided to release Soushi from his contract and Nobara was going to become Ririchiyo's SS. The reason was because since Soushi (and Watanuki) does not have their past memories, it's safer for them that way. The revival/ceremony that killed majority of them in their past lives is approaching, and they decided to stop it this time - without sacrificing anybody. Apparently, there's a curse on Soushi for dying in the previous battle - he's cursed to die the same way in every subsequent life when he reborns. That's why Ririchiyo cancelled their contract to protect him.
Spoiler from someone called ezakey.