  We present microscopic Hamiltonians that gap the Dirac fermions on the
surface of topological crystalline insulators (TCIs) protected by reflection
symmetry and create symmetry-preserving states with Abelian topological orders.
For TCIs with $n = 2$ copies of surface Dirac fermions, we find a fermionic
$\mathbb{Z}_4$ topological order with charge-spin self duality and with a novel
symmetry action on the anyons, whereby reflections with respect to two parallel
planes act differently. We also analyze TCIs with $n = 1$ and $4$ copies of
Dirac fermions, which taken together covers every nontrivial case in the
$\mathbb{Z}_8 \oplus \mathbb{Z}_2$ classification of interacting TCIs. Our work
reveals a fundamental difference between reflection and
charge-conjugation-time-reversal symmetry anomalies.
