A hypergraph is Helly if every family of hyperedges of it, formed by pairwiseintersecting hyperedges, has a common vertex. We consider the concepts ofbipartite-conformal and (colored) bipartite-Helly hypergraphs. In the same way asconformal hypergraphs and Helly hypergraphs are dual concepts, bipartite-conformal andbipartite-Helly hypergraphs are also dual. They are useful for characterizing bicliquematrices and biclique graphs, that is, the incident biclique-vertex incidence matrix andthe intersection graphs of the maximal bicliques of a graph, respectively. These conceptsplay a similar role for the bicliques of a graph, as do clique matrices and clique graphs,for the cliques of the graph. We describe polynomial time algorithms for recognizingbipartite-conformal and bipartite-Helly hypergraphs as well as biclique matrices.