  We discuss (2+1)D topological phases on non-orientable spatial surfaces, such
as M\"obius strip, real projective plane and Klein bottle, etc., which are
obtained by twisting the parent topological phases by their underlying pairty
symmetries through introducing parity defects. We construct the ground states
on arbitrary non-orientable closed manifolds and calculate the ground state
degeneracy. Such degeneracy is shown to be robust against continuous
deformation of the underlying manifold. We also study the action of the mapping
class group on the multiplet of ground states on the Klein bottle. The physical
properties of the topological states on non-orientable surfaces are deeply
related to the parity symmetric anyons which do not have a notion of
orientation in their statistics. The number of ground states on the projective
plane equals the root of the number of distinguishable parity symmetric anyons,
while the ground state degeneracy on the Klein bottle equals the total number
of parity symmetric anyons; In deforming the Klein bottle, the Dehn twist
encodes the topological spins whereas the Y-homeomorphism tells the
particle-hole relation of the parity symmetric anyons.
