  It is shown that the presence of discrete symmetries in Floquet systems
connected to metallic reservoirs imprints a definite parity on the charge and
heat pumping as a function of the reservoir's chemical potential, $\mu$. In
particular, when particle-hole symmetry (PHS) holds, the pumping of charge
(heat) is an odd (even) function of $\mu$. Whereas, if only the product of PHS
and parity symmetry is present, pumping of charge (heat) is even (odd) in
$\mu$. Our results also extend to the presence of other unitary symmetries and
provide a simple criterion for reversing (or maintaining) the direction of the
flow. We illustrate our findings using two variants of the Su-Schrieffer-Heeger
model under a time-periodic perturbation.
