We show that a class of spin models, containing the Ashkin-Teller model, admits a generalized random-cluster (GRC) representation. Moreover, we show that basic properties of the usual representation, such as FKG inequalities and comparison inequalities, still hold for this generalized random-cluster model. Some elementary consequences are given. We also consider the duality transformations in the spin representation and in the GRC model and show that they commute.