Via compression ([18,8]) we write the n-dimensional Chaplygin sphere system as an almost Hamiltonian system on T*SO (n) with internal symmetry group SO (n-1). We show how this symmetry group can be factored out, and pass to the fully reduced system on (a fiber bundle over) T*Sn-1. This approach yields an explicit description of the reduced system in terms of the geometric data involved. Due to this description we can study Hamiltoniz-ability of the system. It turns out that the homogeneous Chaplygin ball, which is not Hamiltonian at the T*SO (n)-level, is Hamiltonian at the T*Sn-1-level. Moreover, the 3-dimensional ball becomes Hamiltonian at the T*S-2-level after time reparametrization, where by we re-prove a result of [4, 5] in symplecto-geometric terms. We also study compression followed by reduction of generalized Chaplygin systems.