Retinal velocity distributions can lead to a percept of three-dimensional (3D) structure (structure-from-motion [SFM]). SFM stimuli are intrinsically ambiguous with regard to depth ordering. A classic example is the orthographic projection of a revolving transparent cylinder, which can be perceived as a 3D cylinder that rotates clockwise and counterclockwise alternately. Prevailing models attribute such bistable percepts to inhibitory connections between neurons that are tuned to opposite motion directions at equal binocular disparities. Cylinder stimuli can yield not only two but as many as four different percepts. Besides the well-documented clockwise and counterclockwise spinning transparent cylinders, observers can also perceive two transparent half-cylinders, either convex or concave, one in front of the other. Observers are able to bias the time during which a percept is present by attending to one or the other percept. We examined this phenomenon quantitatively and found that in standard SFM stimuli, the percept of two convex transparent half-cylinders can occur just as often as the percept of (counter-) clockwise spinning cylinders. So far, however, all interpretations of experimental (neurophysiological) data and all proposed mechanisms for SFM perception have focused solely on the two classical cylinder percepts. Prevailing models cannot explain the existence of the other two percepts. We suggest an alternative model to explain attention-biased multi-stable perception.