Neurons in the visual cortex respond preferentially to edge-like stimuli of a particular orientation. It is a long-standing hypothesis that orientation selectivity arises during development through the activity-dependent refinement of cortical circuitry. Unambiguous evidence for such a process has, however, remained elusive. Here we argue that, if orientation preferences arise through activity-dependent refinement of initially unselective patterns of synaptic connections, this process should leave distinct signatures in the emerging spatial pattern of preferred orientations. Preferred orientations typically change smoothly and progressively across the cortex. This smooth progression is disrupted at the centres of so-called pinwheels, where neurons exhibiting the whole range of orientation preferences are located in close vicinity. Assuming that orientation selectivity develops through a set of rules that we do not specify, we demonstrate mathematically that the spatial density of pinwheels is rigidly constrained by basic symmetry principles. In particular, the spatial density of pinwheels, which emerge when orientation selectivity is first established, is larger than a model-independent minimal value. As a consequence, lower densities, if observed in adult animals, are predicted to develop through the motion and annihilation of pinwheel pairs.