  We study symmetry-enriched topological (SET) phases in 2+1 space-time
dimensions with spatial reflection and/or time-reversal symmetries. We provide
a systematic construction of a wide class of reflection and time-reversal SET
phases in terms of a topological path integral defined on general space-time
manifolds. An important distinguishing feature of different topological phases
with reflection and/or time-reversal symmetry is the value of the path integral
on non-orientable space-time manifolds. We derive a simple general formula for
the path integral on the manifold $\Sigma^2 \times S^1$, where $\Sigma^2$ is a
two-dimensional non-orientable surface and $S^1$ is a circle. This also gives
an expression for the ground state degeneracy of the SET on the surface
$\Sigma^2$ that depends on the reflection symmetry fractionalization class,
generalizing the Verlinde formula for ground state degeneracy on orientable
surfaces. Consistency of the action of the mapping class group on
non-orientable manifolds leads us to a constraint that can detect when a
time-reversal or reflection SET phase is anomalous in (2+1)D and, thus, can
only exist at the surface of a (3+1)D symmetry protected topological (SPT)
state. Given a (2+1)D reflection and/or time-reversal SET phase, we further
derive a general formula that determines which (3+1)D reflection and/or
time-reversal SPT phase hosts the (2+1)D SET phase as its surface termination.
A number of explicit examples are studied in detail.
