McMullen used the fundamental operation of Minkowski sum to construct the polytope algebra of real vector space. In this talk, I will consider the subalgebra generated by deformations of a fixed zonotope and endow it with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. In the particular case of Coxeter arrangements of type A and B, we find striking relations between the corresponding module structure and certain statistics on permutations and signed permutations, respectively. I will explain how these statistics give information on families of polytopes that generate all (type B) generalized permutahedra as signed Minkowski sums.