As the first of the Duodecim, the Mutsukis are a respected family even among elite nobles. They maintain a close relationship with the Amanohokosaka Clan, a bond that was strengthened with Kagura Mutsuki becoming Tenjo Amanohokosaka's personal bodyguard and pupil. Most of their rules and traditions are unknown, but they maintain a generally healthy relationship with all other houses, not seeming judgemental of the Kisaragi and Yayoi families' less orthodox traditions. They care about the heirs of families and have interfered in the assassination attempts of Jin Kisaragi.  According to Jubei, the Mutsukis have produced many strong soldiers. 
They have a branch family, the Kohakus, who specialize in "dirty work" and rarely appear in history books. 
The Mutsukis were against the Novus Orbis Librarium under Izanami's rule. Their dislike was strengthened when they were put in charge of the remains of the 5th Hierarchical City of Ibukido after the Ikaruga Civil War. Kagura spearheaded the rebellion against Izanami to put Homura Amanohokosaka on the NOL's throne and united the Duodecim for the cause.
BlazBlue: Central Fiction Library Entry
One of the families of the Duodecim. 'Mutsuki' means 'the first moon'.
The current head of the family is Kagura Mutsuki. They object to the Imperator's actions, and oppose the Librarium. 
- BlazBlue: Remix Heart, Chapter 19
- BlazBlue: Central Fiction, Arcade Mode, Jubei Victory Quote Vs Kagura Mutsuki, Act III
- BlazBlue: Chrono Phantasma Extend, Story Mode, Kagura "Determination", Episodes 1 - 2
- BlazBlue: Central Fiction, Library Mode, Organization #01