  We define a moduli space of rational curves with finite-order automorphism
and weighted orbits, and we prove that the combinatorics of its boundary strata
are encoded by a particular polytopal complex that also captures the algebraic
structure of a complex reflection group acting on the moduli space. This
generalizes the situation for Losev-Manin's moduli space of curves (whose
boundary strata are encoded by the permutohedron and related to the symmetric
group) as well as the situation for Batyrev-Blume's moduli space of curves with
involution, and it extends that work beyond the toric context.
