Orientation illusions occur when the inducing figure is a line or grating (the tilt illusion) or a square outline frame (the rod-and-frame illusion). In the range of inducing figure tilts between vertical and horizontal, the tilt illusion describes one cycle of positive (direct) and negative (indirect) effects but the rod-and-frame illusion describes two such cycles. In two experiments, angular functions of illusions were measured with the six possible inducing figures which result when two of the four sides of a square inducing frame are deleted. As expected, the parallel-side frame amputations induced angular functions similar to the tilt illusion and these functions differed from those induced by the orthogonal-sided amputations. In agreement with previous findings on the nonadditivity of tilt illusions, the sum of angular functions induced by frame amputations, which together form a complete frame, were not always equivalent to the angular function induced by a complete frame, and there were asymmetries in the data fro which neither of two simple hypotheses could adequately account. The discussion focuses upon properties of inducing figures which psychophysical hypotheses might need to consider in order to account for the shapes of angular functions of orientation illusions and, in particular,a distinction is drawn between the global orientation of the inducing figure and the orientations of it (local) component features. It is suggested that it might be fruitful if the tilt illusion and the rod-and-frame illusion were conceived of as illusions resulting from inducing figures composed of all or part of n gratings of spatial frequency fn intersecting at angles of 180 degrees/n.