Two-phase random textures abound in a host of contexts, includ- ing porous and composite media, ecological structures, biological media, and astrophysical structures. Questions surrounding the spatial structure of such textures continue to pose many theoret- ical challenges. For example, can two-point correlation functions be identified that can be manageably measured and yet reflect nontrivial higher-order structural information about the textures? We present a solution to this question by probing the information content of the widest class of different types of two-point func- tions examined to date using inverse “reconstruction” techniques. This enables us to show that a superior descriptor is the two-point cluster function C₂(r), which is sensitive to topological connected- ness information. We demonstrate the utility of C₂(r) by accurately reconstructing textures drawn from materials science, cosmology, and granular media, among other examples. Our work suggests a theoretical pathway to predict the bulk physical properties of ran- dom textures and that also has important ramifications for atomic and molecular systems.