  Two ideal polygons, $(p_1,\ldots,p_n)$ and $(q_1,\ldots,q_n)$, in the
hyperbolic plane or in hyperbolic space are said to be $\alpha$-related if the
cross-ratio $[p_i,p_{i+1},q_i,q_{i+1}] = \alpha$ for all $i$ (the vertices lie
on the projective line, real or complex, respectively). For example, if $\alpha
= -1$, the respective sides of the two polygons are orthogonal. This relation
extends to twisted ideal polygons, that is, polygons with monodromy, and it
descends to the moduli space of M\"obius-equivalent polygons. We prove that
this relation, which is, generically, a 2-2 map, is completely integrable in
the sense of Liouville. We describe integrals and invariant Poisson structures,
and show that these relations, with different values of the constants $\alpha$,
commute, in an appropriate sense. We investigate the case of small-gons,
describe the exceptional ideal polygons, that possess infinitely many
$\alpha$-related polygons, and study the ideal polygons that are
$\alpha$-related to themselves (with a cyclic shift of the indices).
