  A one dimensional time reversal symmetric topological superconductor
(symmetry class DIII) features a single Kramers pair of Majorana bound states
at each of its ends. These holographic quasiparticles are non-Abelian anyons
that obey Ising-type braiding statistics. In the special case where an
additional U(1) spin rotation symmetry is present, this state can be understood
as two copies of a Majorana wire in symmetry class D, one copy for each spin
block. We present a manifestly gauge invariant construction of the topological
invariant for the generic case, i.e., in the absence of any additional
symmetries like spin rotation symmetry. Furthermore, we show how the presence
of inversion symmetry simplifies the calcuation of the topological invariant.
The proposed scheme is suitable for the classification of both interacting and
disordered systems and allows for a straightforward numerical evaluation of the
invariant since it does not rely on fixing a continuous phase relation between
Bloch functions. Finally, we apply our method to compute the topological phase
diagram of a Rashba wire with competing s-wave and p-wave superconducting
pairing terms.
