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In this paper, Slepian-Wolf coding with a mismatched decoding metric is studied. Two different dualities between Slepian-Wolf coding and channel coding under mismatched decoding are established. These two dualities provide a systematic framework for comparing linear Slepian-Wolf codes, nonlinear Slepian-Wolf codes, and variable-rate Slepian-Wolf codes. In contrast with the fact that linear codes suffice to achieve the Slepian-Wolf limit under matched decoding, the minimum rate achievable with nonlinear Slepian-Wolf codes under mismatched decoding can be strictly lower than that achievable with linear Slepian-Wolf codes.