We study directed transport in classical and quantum area-preserving maps, periodic in space and momentum. On the classical level, we show that a sum rule excludes directed transport of the entire phase space, leaving only the possibility of transport in (dynamically defined) subsets, such as regular islands or chaotic areas. As a working example, we construct a mapping with a mixed phase space where both the regular and the chaotic components support directed currents, but with opposite sign. The corresponding quantum system shows transport of similar strength, associated to the same subsets of phase space as in the classical map.