A mask of a face rotated about its vertical axis of symmetry can appear to oscillate rather than rotate. Do stimulus features (e.g., shape) or cognitive factors (e.g., differential familiarity with convex and concave views of faces) explain this new illusion? In Experiment 1, differential familiarity was varied across stimuli by using familiar and unfamiliar objects rotating at 4 rpm and within stimuli by showing the objects upright and inverted. True motion was seen more with unfamiliar objects than with familiar objects and more with an inverted mask than with an upright mask. The results of Experiment 2, which was done with static views, suggest that the upright and inverted masks present similar structure to the visual system. In Experiment 3, the objects were shown rotating at 8 rpm; the results are similar to those of Experiment 1. These experiments favor a differential familiarity account of this illusory motion. Cognitive constraints on perceived motion and perceived rigidity are discussed.