Binocular disparity, the differential angular separation between pairs of image points in the two eyes, is the well-recognized basis for binocular distance perception. Without denying disparity's role in perceiving depth, we describe two perceptual phenomena, which indicate that a wider view of binocular vision is warranted. First, we show that disparity can play a critical role in two-dimensional perception by determining whether separate image fragments should be grouped as part of a single surface or segregated as parts of separate surfaces. Second, we show that stereoscopic vision is not limited to the registration and interpretation of binocular disparity but that it relies on half-occluded points, visible to one eye and not the other, to determine the layout and transparency of surfaces. Because these half-visible points are coded by neurons carrying eye-of-origin information, we suggest that the perception of these surface properties depends on neural activity available at visual cortical area V1.