  Topological phases are enriched in non-equilibrium open systems effectively
described by non-Hermitian Hamiltonians. While several properties unique to
non-Hermitian topological systems were uncovered, the fundamental role of
symmetry in non-Hermitian physics has yet to be fully understood, and it has
remained unclear how symmetry protects non-Hermitian topological phases. Here
we show that two fundamental anti-unitary symmetries, time-reversal and
particle-hole symmetries, are topologically equivalent in the complex energy
plane and hence unified in non-Hermitian physics. A striking consequence of
this symmetry unification is the emergence of unique non-equilibrium
topological phases that have no counterparts in Hermitian systems. We
illustrate this by presenting a non-Hermitian counterpart of the Majorana chain
in an insulator with time-reversal symmetry and that of the quantum spin Hall
insulator in a superconductor with particle-hole symmetry. Our work establishes
a fundamental symmetry principle in non-Hermitian physics and paves the way
towards a unified framework for non-equilibrium topological phases.
