  We investigate the role of symmetries in determining the random matrix class
describing quantum thermalization in a periodically driven many body quantum
system. Using a combination of analytical arguments and numerical exact
diagonalization, we establish that a periodically driven `Floquet' system can
be in a different random matrix class to the instantaneous Hamiltonian. A
periodically driven system can thermalize even when the instantaneous
Hamiltonian is integrable. A Floquet system that thermalizes in general can
display integrable behavior at commensurate driving frequencies. When the
instantaneous Hamiltonian and Floquet operator both thermalize, the Floquet
problem can be in the unitary class while the instantaneous Hamiltonian is
always in the orthogonal class, and vice versa. We extract general principles
regarding when a Floquet problem can thermalize to a different symmetry class
to the instantaneous Hamiltonian. A (finite-sized) Floquet system can even
display crossovers between different random matrix classes as a function of
driving frequency.
