  We investigate, within Floquet theory, topological phases in the
out-of-equilibrium system that consists of fermions in a circularly shaken
honeycomb optical lattice. We concentrate on the intermediate regime, in which
the shaking frequency is of the same order of magnitude as the band width, such
that adjacent Floquet bands start to overlap, creating a hierarchy of band
inversions. It is shown that two-phonon resonances provide a topological phase
that can be described within the Bernevig-Hughes-Zhang model of HgTe quantum
wells. This allows for an understanding of out-of-equilibrium topological
phases in terms of simple band inversions, similar to equilibrium systems.
