  The effect of interactions on topological insulators and superconductors
remains, to a large extent, an open problem. Here, we describe a framework for
classifying phases of one-dimensional interacting fermions, focusing on
spinless fermions with time-reversal symmetry and particle number parity
conservation, using concepts of entanglement. In agreement with an example
presented by Fidkowski \emph{et. al.} (Phys. Rev. B 81, 134509 (2010)), we find
that in the presence of interactions there are only eight distinct phases,
which obey a $\mathbb{Z}_8$ group structure. This is in contrast to the
$\mathbb{Z}$ classification in the non-interacting case. Each of these eight
phases is characterized by a unique set of bulk invariants, related to the
transformation laws of its entanglement (Schmidt) eigenstates under symmetry
operations, and has a characteristic degeneracy of its entanglement levels. If
translational symmetry is present, the number of distinct phases increases to
16.
