  It is well known that the Eulerian polynomial is the Hilbert series of the
cohomology of the permutohedral variety. We answer a question of Stembridge on
finding a geometric explanation of the \emph{permutation representation} this
cohomology carries. Our explanation involves an $\mathfrak{S}_n$-equivariant
bijection between a basis for the Chow ring of the Boolean matroid and codes
introduced by Stembridge. There are analogous results for the stellohedral
variety. We provide a geometric explanation of the permutation representation
that its cohomology carries. This involves the augmented Chow ring of a matroid
introduced by Braden, Huh, Matherne, Proudfoot and Wang. Along the way, we also
obtain some new results on augmented Chow rings.
