  Involution words are variations of reduced words for twisted involutions in
Coxeter groups. They arise naturally in the study of the Bruhat order, of
certain Iwahori-Hecke algebra modules, and of orbit closures in flag varieties.
Specifically, to any twisted involutions $x$, $y$ in a Coxeter group $W$ with
automorphism $*$, we associate a set of involution words
$\hat{\mathcal{R}}_*(x,y)$. This set is the disjoint union of the reduced words
of a set of group elements $\mathcal{A}_*(x,y)$, which we call the atoms of $y$
relative to $x$. The atoms, in turn, are contained in a larger set
$\mathcal{B}_*(x,y) \subset W$ with a similar definition, whose elements we
refer to as Hecke atoms. Our main results concern some interesting properties
of the sets $\hat{\mathcal{R}}_*(x,y)$ and $\mathcal{A}_*(x,y) \subset
\mathcal{B}_*(x,y)$. For finite Coxeter groups we prove that
$\mathcal{A}_*(1,y)$ consists of exactly the minimal-length elements $w \in W$
such that $w^* y \leq w$ in Bruhat order, and conjecture a more general
property for arbitrary Coxeter groups. In type $A$, we describe a simple set of
conditions characterizing the sets $\mathcal{A}_*(x,y)$ for all involutions
$x,y \in S_n$, giving a common generalization of three recent theorems of Can,
Joyce, and Wyser. We show that the atoms of a fixed involution in the symmetric
group (relative to $x=1$) naturally form a graded poset, while the Hecke atoms
surprisingly form an equivalence class under the "Chinese relation" studied by
Cassaigne, Espie, et al. These facts allow us to recover a recent theorem of Hu
and Zhang describing a set of "braid relations" spanning the involution words
of any self-inverse permutation. We prove a generalization of this result
giving an analogue of Matsumoto's theorem for involution words in arbitrary
Coxeter groups.
