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We study the steady-state behaviors of a typical optomechanical cavity coupled to cold Rydberg atoms with dipole-dipole interactions. The interacting atoms are described as one superatom of three collective states in a ladder configuration in the limit of a strong dipole blockade and a weak cavity field. We find that this hybrid system exhibits phenomena of conditional duality and nonlinear bistability in terms of mirror displacement, number of cavity photons, and Rydberg population, depending on the detuning of the cavity field, the strength of the optical driving field, and the number of cold atoms. It is of particular interest that the two branches of relevant curves may intersect to yield a nontrivial duality and bistability. Such correlated optical, mechanical, and atomic responses arise from the efficient feedback between atom-light and optomechanical interactions and have realistic applications, e.g., in realizing accurate optomechanical detection or attaining deterministic single photons. [ABSTRACT FROM AUTHOR]