  Via compression ([11, 7]) 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) $T^*S^{n-1}$. This
approach yields an explicit description of the reduced system in terms of the
geometric data involved. Due to this description we can study Hamiltonizability
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 $T^*S^{n-1}$-level.
Moreover, the 3-dimensional ball becomes Hamiltonian at the $T^*S^2$-level
after time reparametrization, whereby we re-prove a result of [4, 5] in
symplecto-geometric terms. We also study compression followed by reduction of
generalized Chaplygin systems.
