One of the main theoretical challenges of vision science is to explain how the visual system interpolates missing structure. Two forms of visual completion have been distinguished on the basis of the phenomenological states that they induce. "Modal" completion refers to the formation of visible surfaces and/or contours in image regions where these properties are not specified locally. "Amodal" completion refers to the perceived unity of objects that are partially obscured by occluding surfaces. Although these two forms of completion elicit very different phenomenological states, it has been argued that a common mechanism underlies modal and amodal boundary and surface interpolation (the "identity hypothesis"; Kellman & Shipley, 1991; Kellman, 2001). Here, we provide new data, demonstrations, and theoretical principles that challenge this view. We show that modal boundary and surface completion processes exhibit a strong dependence on the prevailing luminance relationships of a scene, whereas amodal completion processes do not. We also demonstrate that the shape of interpolated contours can change when a figure undergoes a transition from a modal to an amodal appearance, in direct contrast to the identity hypothesis. We argue that these and previous results demonstrate that modal and amodal completion do not result from a common interpolation mechanism.