  We consider an effect of the discrete spatial symmetries and magnetic field
on the adiabatic charge pumping in mesoscopic systems. In general case, there
is no symmetry of the pumped charge with respect to the inversion of magnetic
field Q(B) \neq Q(-B). We find that the reflection symmetries give rise to
relations Q(B)=Q(-B) or Q(B)=-Q(-B) depending on the orientation of the
reflection axis. In presence of the center of inversion, Q(B) = 0. Additional
symmetries may arise in the case of bilinear pumping.
