  Let $W\ltimes L$ be an irreducible affine Weyl group with Coxeter complex
$\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the
translation subgroup. The Steinberg torus is the Boolean cell complex obtained
by taking the quotient of $\Sigma$ by the lattice $L$. We show that the
ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face
deleted) are generating functions over $W$ for a descent-like statistic first
studied by Cellini. We also show that the ordinary $h$-polynomial has a
nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In
the classical cases, we also provide expansions, identities, and generating
functions for the $h$-polynomials of Steinberg tori.
