Two ideal polygons, \( (p_1,\dots,p_n) \) and \( (q_1,\dots,q_n) \), 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 Möbius-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).
