We introduce a novel statistical way of analyzing the projected mass distribution in galaxy lenses based solely on the angular distribution of images in four-image systems ("quads") around the lens center. The method requires the knowledge of the lens center location, but the images' distances from the lens center are not used at all. If the images of a quad are numbered in order of arrival time, θ1 through θ4, and θij is the angle between images i and j, then we define the "bisector" plane whose axes are linear combinations of θ23 and θ14. The bisector plane of a given lens contains all the quads produced by the lens. We show empirically that all two-fold symmetric lenses with convex, i.e., nonwavy or petal-like, isodensity contours are identical in the bisector plane of their quads. We also study lenses with twisting isodensity contours, lumpy substructure, etc. Our results suggest that to reproduce the general characteristics of the observed quad population, kiloparsec-scale substructure must be a common feature of galaxy lenses.