  We define and compute many-body topological invariants of interacting
fermionic symmetry-protected topological phases, protected by an
orientation-reversing symmetry, such as time-reversal or reflection symmetry.
The topological invariants are given by partition functions obtained by a path
integral on unoriented spacetime which, as we show, can be computed for a given
ground state wave function by considering a non-local operation, "partial"
reflection or transpose. As an application of our scheme, we study the
$\mathbb{Z}_8$ and $\mathbb{Z}_{16}$ classification of topological
superconductors in one and three dimensions.
