Abstract. This paper is a sequel to [4]. We establish the minimal basis theory for the centralizers of parabolic subalgebras of Iwahori--Hecke algebras associated to finite Coxeter groups of any type, generalizing the approach introduced in [3] from centres to the centralizer case. As a pre-requisite, we prove a reducibility property in the twisted [MATHEMATICAL FORMULA]-conjugacy classes in finite Coxeter groups, which is a generalization of results in [7] and [4].