  Many geometric structures associated to surface groups can be encoded in
terms of invariant cross ratios on their circle at infinity; examples include
points of Teichm\"uller space, Hitchin representations and geodesic currents.
We add to this picture by studying cubulations of arbitrary Gromov hyperbolic
groups $G$. Under weak assumptions, we show that the space of cubulations of
$G$ naturally injects into the space of $G$-invariant cross ratios on the
Gromov boundary $\partial_{\infty}G$.
  A consequence of our results is that essential, hyperplane-essential
cubulations of hyperbolic groups are length-spectrum rigid, i.e. they are fully
determined by their length function. This is the optimal length-spectrum
rigidity result for cubulations of hyperbolic groups, as we demonstrate with
some examples. In the hyperbolic setting, this constitutes a strong improvement
on our previous work in arXiv:1903.02447.
  Along the way, we describe the relationship between the Roller boundary of a
${\rm CAT(0)}$ cube complex, its Gromov boundary and - in the non-hyperbolic
case - the contracting boundary of Charney and Sultan.
  All our results hold for cube complexes with variable edge lengths.
