Twisted identities in Coxeter groups

Given a Coxeter system (W,S) equipped with an involutive automorphism T, the set of twisted identities is i(T) = {T(w)^{-1}w : w \in W}. We point out how i(T) shows up in several contexts and prove that if there is no s \in S such that sT(s) is of odd order greater than 1, then the Bruhat order on i(T) is a graded poset with rank function r given by halving the Coxeter length. Under the same condition, it is shown that the order complexes of the open intervals either are PL spheres or acyclic over the integers. In the general case, contractibility is shown for certain classes of intervals. Furthermore, we demonstrate that sometimes these posets are not graded. For the Poincar\'e series of i(T), i.e. its generating function with respect to r, a factorisation phenomenon is discussed.