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In this paper, it is shown that each Slepian-Wolf coding problem is related to a dual channel coding problem in the sense that the sphere packing exponents, random coding exponents, and correct decoding exponents in these two problems are mirror-symmetrical to each other. This mirror symmetry is interpreted as a manifestation of the linear codebook-level duality between Slepian-Wolf coding and channel coding. Furthermore, this duality, in conjunction with a systematic analysis of the expurgated exponents, reveals that nonlinear Slepian-Wolf codes can strictly outperform linear Slepian-Wolf codes in terms of rate-error tradeoff at high rates. The linear codebook-level duality is also established for general sources and channels.