  We classify the topology of bands defined by the energy states of quantum
systems with scale separation between slow and fast degrees of freedom,
invariant under fermionic time reversal. Classical phase space transforms
differently from momentum space under time reversal, and as a consequence the
topology of adiabatic bands is different from that of Bloch bands. We show that
bands defined over a two-dimensional phase space are classified by the Chern
number, whose parity must be equal to the parity of the band rank. Even-rank
bands are equivalently classified by the Kane-Mele index, an integer equal to
one half the Chern number.
