  Given an involutive automorphism $\theta$ of a Coxeter system $(W,S)$, let
$\mathfrak{I}(\theta) \subseteq W$ 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 $\underline{S}$-expressions (also known as admissible
sequences, reduced $I_\theta$-expressions, or involution words) for any given
$w \in \mathfrak{I}(\theta)$. 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.

Combinatorics