We prove multiple generalizations of Fan's combinatorial labeling result for sphere triangulations. This can be seen as a comprehensive extension of the Borsuk--Ulam theorem. In typical applications, the Borsuk--Ulam theorem gives complexity bounds in a suitable sense, whereas our extension additionally provides insight into the structure of objects satisfying the complexity bound. This structure is governed by order types of finite point sets in Euclidean space and more generally by the intersection combinatorics of faces under continuous maps from the simplex. We develop some of those applications for sphere coverings, Kneser-type colorings, Hall-type results for hypergraphs, and hyperplane mass partitions, among other consequences. We provide a new proof of the topological Hall theorem and extend it into a result that simultaneously generalizes hypergraph Hall theorems and topological lower bounds for chromatic numbers.