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In this paper, we develop a bipartite network framework to study the robustness of interdependent hypergraphs. From such a perspective, nodes and hyperedges of a hypergraph are equivalent to each other, a property that largely simplifies their mathematical treatment. We develop a general percolation theory based on this representation and apply it to study the robustness of interdependent hypergraphs against random damage, which we verify with numerical simulations. We analyze a variety of interacting patterns, from heterogeneous to correlated hyperstructures, and from full- to partial-dependency couplings between an arbitrary number of hypergraphs, and characterize their structural stability via their phase diagrams. Given its generality, we expect that our framework will provide useful insights for the development of more realistic venues to characterize cascading failures in interdependent higher-order systems.