Danilov-Koshevoy, Postnikov, and Ardila-Benedetti-Doker taught us that any generalized permutahedron is a signed Minkowski sum of the faces of the standard simplex. In other words, these faces correspond to a maximal linearly independent collection of rays in the deformation cone of the permutahedron. In contrast, Ardila-Castillo-Eur-Postnikov observed that the faces of the cross-polytope only span a subspace of roughly half the dimension of the deformation cone of the type B permutahedron. In this talk, we use McMullen's polytope algebra to help explain this phenomenon.