It is well known that the Eulerian polynomial is the Hilbert series of the cohomology of the permutahedral variety. In this talk, we answer a question of Stembridge on finding a permutation basis for the permutation representation this cohomology carries. Our method involves an S_n-equivariant bijection between a basis for the Chow ring of the Boolean matroid and codes introduced by Stembridge. There is a parallel binomial Eulerian story related to the stellahedral variety. We also find a permutation basis for the permutation representation on the cohomology of the stellahedral variety. 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.