  We classify the gapped phases of Z_N parafermions in one dimension and
construct a representative of each phase. Even in the absence of additional
symmetries besides parafermionic parity, parafermions may be realized in a
variety of phases, one for each divisor n of N. The phases can be characterized
by spontaneous symmetry breaking, topology, or a mixture of the two. Purely
topological phases arise if n is a unitary divisor, i.e. if n and N/n are
co-prime. Our analysis is based on the explicit realization of all symmetry
broken gapped phases in the dual Z_N-invariant quantum spin chains.
