  We extend the single-particle topological classification of insulators and
superconductors to include systems in which disorder preserves average
reflection symmetry. We show that the topological group structure of bulk
Hamiltonians and topological defects is exponentially extended when this
additional condition is met, and examine some of its physical consequences.
Those include localization-delocalization transitions between topological
phases with the same boundary conductance, as well as gapless topological
defects stabilized by average reflection symmetry.
