  Recently it has been proposed that a unitary topological mirror symmetry can
stabilize multiple zero energy Majorana fermion modes in one dimensional (1D)
time reversal (TR) invariant topological superconductors. Here we establish an
exact equivalence between 1D "topological mirror superconductivity" and chiral
topological superconductivity in BDI class which can also stabilize multiple
Majorana-Kramers pairs in 1D TR-invariant topological superconductors. The
equivalence proves that topological mirror superconductivity can be understood
as chiral superconductivity in the BDI symmetry class co-existing with
time-reversal symmetry. Furthermore, we show that the mirror Berry phase
coincides with the chiral winding invariant of the BDI symmetry class, which is
independent of the presence of the time-reversal symmetry. Thus, the
time-reversal invariant topological mirror superconducting state may be viewed
as a special case of the BDI symmetry class in the well-known Altland-Zirnbauer
periodic table of free fermionic phases. We illustrate the results with the
examples of 1D spin-orbit coupled quantum wires in the presence of nodeless
s_{\pm} superconductivity and the recently discussed experimental system of
ferromagnetic atom (Fe) chains embedded on a lead (Pb) superconductor.
