A bicolored rectangular family BRF is a collection of all axis-parallel rectangles contained in a given region Z of the plane formed by selecting a bottom-left corner from a set A and an upper-right corner from a set B. We prove that the maximum independent set and the minimum hitting set of a BRF have the same cardinality and devise polynomial time algorithms to compute both. As a direct consequence, we obtain the first polynomial time algorithm to compute minimum biclique covers, maximum cross-free matchings and jump numbers in a class of bipartite graphs that significantly extends convex bipartite graphs and interval bigraphs. We also establish several connections between our work and other seemingly unrelated problems. Furthermore, when the bicolored rectangular family is weighted, we show that the problem of finding the maximum weight of an independent set is NP-hard, and provide efficient algorithms to solve it on certain subclasses.