  Wigner's quantum-mechanical classification of particle-types in terms of
irreducible representations of the Poincar\'e group has a classical analogue,
which we extend in this paper. We study the compactness properties of the
resulting phase spaces at fixed energy, and show that in order for a classical
massless particle to be physically sensible, its phase space must feature a
classical-particle counterpart of electromagnetic gauge invariance. By
examining the connection between massless and massive particles in the massless
limit, we also derive a classical-particle version of the Higgs mechanism.
