Given an involutive automorphism theta of a Coxeter system (W, S), let J(theta) subset of W denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced S-expressions (also known as admissible sequences, reduced I-theta-expressions, or involution words) for any given w is an element of J(theta). This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg. (C) 2018 Elsevier Inc. All rights reserved.