  Recent works have demonstrated that the Floquet-Bloch bands of
periodically-driven systems feature a richer topological structure than their
non-driven counterparts. The additional structure in the driven case arises
from the periodicity of quasienergy, the energy-like quantity that defines the
spectrum of a periodically-driven system. Here we develop a new paradigm for
the topological classification of Floquet-Bloch bands, based on the
time-dependent spectrum of the driven system's evolution operator throughout
one driving period. Specifically, we show that this spectrum may host
topologically-protected degeneracies at intermediate times, which control the
topology of the Floquet bands of the full driving cycle. This approach provides
a natural framework for incorporating the role of symmetries, enabling a
unified and complete classification of Floquet-Bloch bands and yielding new
insight into the topological features that distinguish driven and non-driven
systems.
