We study the duality between channel coding and Slepian-Wolf coding in the linear coding framework. We show how a code (both its encoder and decoder) for a symmetric channel coding problem can be used to design a code for a general Slepian-Wolf problem. Conversely, we show how a code for a symmetric Slepian-Wolf problem can be used to design a code for a general channel coding problem. The exact relations between the rates and the probability of errors of the two codes are established.