  We classify interacting topological insulators and superconductors with
order-two crystal symmetries (reflection and twofold rotation), focusing on the
case where interactions reduce the noninteracting classification. We find that
the free-fermion $\mathbb{Z}_2$ classifications are stable against quartic
contact interactions, whereas the $\mathbb{Z}$ classifications reduce to
$\mathbb{Z}_N$, where $N$ depends on the symmetry class and the dimension $d$.
These results are derived using a quantum nonlinear $\sigma$ model (QNLSM) that
describes the effects of the quartic interactions on the boundary modes of the
crystalline topological phases. We use Clifford algebra extensions to derive
the target spaces of these QNLSMs in a unified way. The reduction pattern of
the free-fermion classification then follows from the presence or absence of
topological terms in the QNLSMs, which is determined by the homotopy group of
the target spaces. We show that this derivation can be performed using either a
complex fermion or a real Majorana representation of the crystalline
topological phases and demonstrate that these two representations give
consistent results. To illustrate the breakdown of the noninteracting
classification we present examples of crystalline topological insulators and
superconductors in dimensions one, two, and three, whose surfaces modes are
unstable against interactions. For the three-dimensional example, we show that
the reduction pattern obtained by the QNLSM method agrees with the one inferred
from the stability analysis of the boundary modes using bosonization.
