  Twisting symmetries provides an efficient method to diagnose
symmetry-protected topological (SPT) phases. In this paper, edge theories of
(2+1)-dimensional topological phases protected by reflection as well as other
symmetries are studied by twisting reflection symmetry, which effectively puts
the edge theories on an unoriented spacetime, such as the Klein bottle. A key
technical step taken in this paper is the use of the so-called cross-cap
states, which encode entirely the unoriented nature of spacetime, and can be
obtained by rearranging the spacetime geometry and exchanging the role of space
and time coordinates. When the system is in a non-trivial SPT phase, we find
that the corresponding cross-cap state is non-invariant under the action of the
symmetries of the SPT phase, but acquires an anomalous phase. This anomalous
phase, with a proper definition of a reference state, on which symmetry acts
trivially, reproduces the known classification of (2+1)-dimensional bosonic and
fermionic SPT phases protected by reflection symmetry, including in particular
the Z_8 classification of topological crystalline superconductors protected by
reflection and time-reversal symmetries.
