Toric varieties modulo reflections

Let \(W\) be a finite group generated by reflections of a lattice \(M\). If a lattice polytope \(P \subset M \otimes_{\mathbb Z}\mathbb R\) is preserved by \(W\), then we show that the quotient of the projective toric variety \(X_P\) by \(W\) is isomorphic to the toric variety \(X_{P \cap D}\), where \(D\) is a fundamental domain for the action of \(W\). This answers a question of Horiguchi-Masuda-Shareshian-Song, and recovers results of Blume, of the second author, and of Gui-Hu-Liu.