  While free fermion topological crystalline insulators have been largely
classified, the analogous problem in the strongly interacting case has been
only partially solved. In this paper, we develop a characterization and
classification of interacting, invertible fermionic topological phases in (2+1)
dimensions with charge conservation, discrete magnetic translation and $M$-fold
point group rotation symmetries, which form the group $G_f = \text{U}(1)^f
\times_{\phi} [\mathbb{Z}^2\rtimes \mathbb{Z}_M]$ for $M=1,2,3,4,6$. $\phi$ is
the magnetic flux per unit cell. We derive a topological response theory in
terms of background crystalline gauge fields, which gives a complete
classification of different phases and a physical characterization in terms of
quantized response to symmetry defects. We then derive the same classification
in terms of a set of real space invariants $\{\Theta_{\text{o}}^\pm\}$ that can
be obtained from ground state expectation values of suitable partial rotation
operators. We explicitly relate these real space invariants to the quantized
coefficients in the topological response theory, and find the dependence of the
invariants on the chiral central charge $c_-$ of the invertible phase. Finally,
when $\phi = 0$ we derive an explicit map between the free and interacting
classifications.
