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In this article, we study capacitated network design problems. We unify and extend polyhedral results for directed, bidirected, and undirected link capacity models. Valid inequalities based on a network cut are known to be strong in several special cases. We show that regardless of the link model, facets of the polyhedra associated with such a cut translate to facets of the original network design polyhedra if the two subgraphs defined by the network cut are (strongly) connected. Our investigation of the facial structure of the cutset polyhedra allows to complement existing polyhedral results for the three variants by presenting facet‐defining flow‐cutset inequalities in a unifying way. In addition, we present a new class of facet‐defining inequalities, showing as well that flow‐cutset inequalities alone do not suffice to give a complete description for single‐commodity, single‐module cutset polyhedra in the bidirected and …