  If $G$ is a group acting geometrically on a CAT(0) cube complex $X$ and if $g
\in G$ is an infinite-order element, we show that exactly one of the following
situations occurs: (i) $g$ defines a rank-one isometry of $X$; (ii) the stable
centraliser $SC_G(g)= \{ h \in G \mid \exists n \geq 1, [h,g^n]=1 \}$ of $g$ is
not virtually cyclic; (iii) $\mathrm{Fix}_Y(g^n)$ is finite for every $n \geq
1$ and the sequence $(\mathrm{Fix}_Y(g^n))$ takes infinitely many values, where
$Y$ is a cubical component of the Roller boundary of $X$ which contains an
endpoint of an axis of $g$. We also show that (iii) cannot occur in several
cases, providing a purely algebraic characterisation of rank-one isometries.
