We develop a formalism to describe the scattering of dark matter (DM) particles by electrons bound in crystals for a general form of the underlying DM-electron interaction. Such a description is relevant for direct-detection experiments of DM particles lighter than a nucleon, which might be observed in operating DM experiments via electron excitations in semiconductor crystal detectors. Our formalism is based on an effective theory approach to general nonrelativistic DM-electron interactions, including the anapole, and magnetic and electric dipole couplings, combined with crystal response functions defined in terms of electron wave function overlap integrals. Our main finding is that, for the usual simplification of the velocity integral, the rate of DM-induced electronic transitions in a semiconductor material depends on at most five independent crystal response functions four of which are distinct from the usual scalar response. We identify these crystal responses and evaluate them using density functional theory for crystalline silicon and germanium, which are used in operating DMdirect-detection experiments. Our calculations allow us to set 90% confidence level limits on the strength of DM-electron interactions from data reported by the SENSEI and EDELWEISS experiments. The crystal response functions discovered in this paper encode properties of crystalline solids that do not interact with conventional experimental probes, suggesting the use of the DM wind as a probe to reveal new kinds of hidden order in materials.