Large complex networks are often organized into groups or communities. In this paper, we introduce and investigate a generative model of network evolution that reproduces all four pairwise community types that exist in directed networks: assortative, core-periphery, disassortative, and the newly introduced source-basin type. We fix the number of nodes and the community membership of each node, allowing node connectivity to change through rewiring mechanisms that depend on the community membership of the involved nodes. We determine the dependence of the community relationship on the model parameters using a mean-field solution. It reveals that a difference in the swap probabilities of the two communities is a necessary condition to obtain a core-periphery relationship and that a difference in the average in-degree of the communities is a necessary condition for a source-basin relationship. More generally, our analysis reveals multiple possible scenarios for the transition between the different structure types, and sheds light on the mechanisms underlying the observation of the different types of communities in network data.