Symmetry-protected zero modes in quantum physics traditionally arise in the context of ground states of many-body Hamiltonians, and are especially prominent in noninteracting problems. In this talk, I will discuss how protected zero modes can arise in otherwise generic interacting quantum many-body systems with spectral reflection symmetries. These many-body zero modes exist at finite energy densities above the ground state, and can even appear when conservation of energy is disposed of altogether; as such, they give rise to a notion of protected many-body degeneracy at "infinite temperature." I will discuss index theorems that protect these zero modes, and give examples of many-body systems that host them. In particular, I will introduce a class of nonintegrable spin chains that includes the so-called "Fibonacci chain" that was recently realized in Rydberg-atom quantum simulators, and show how the presence of zero modes can be definitively inferred from the long-time dynamics of certain operators. I will also show how this situation generalizes to periodically-driven (or "Floquet") quantum systems, where protected degenerate states can give rise to subharmonic response reminiscent of, but distinct from, that of discrete time crystals.