  We show that time-reflection symmetry in periodically driven (Floquet)
quantum systems enables an inherently nonequilibrium phenomenon structurally
similar to quantum-mechanical sypersymmetry. In particular, we find Floquet
analogues of the Witten index that place lower bounds on the degeneracies of
states with quasienergies $0$ and $\pi$. Moreover, we show that in some cases
time reflection symmetry can also interchange fermions and bosons, leading to
fermion/boson pairs with opposite quasienergy. We provide a simple class of
disordered, interacting, and ergodic Floquet models with an exponentially large
number of states at quasienergies $0$ and $\pi$, which are robust as long as
the time-reflection symmetry is preserved. Floquet supersymmetry manifests
itself in the evolution of certain local observables as a period-doubling
effect with dramatic finite-size scaling, providing a clear signature for
experiments.
