Given a clutter L , we associate with it the covering polyhedron Q L , that is the dominant of the convex hull of incidence vectors of all the covers of L . In this paper we describe a binary composition operation, the anti-join, which combines a pair of clutters L 1 and L 2 to give a new clutter L . The anti-join operation is, in some sense, the “dual” of the join operation, introduced by Cunningham [10]. In fact, it has the property that the blocker of the clutter L obtained by joining two clutters L 1 and L 2, is the anti-join of the blockers of L 1 and L 2. For such an operation we show how the linear descriptions of the polyhedra Q( L 1) and Q( L 2) have to be combined to produce a linear description of the polyhedron Q L . Moreover, given a set F ⊆ {λ: 0⩽λ⩽1} such that {0, 1} ⊆ F and aϵ F if and only if (1−a)ϵ F , we define the F -property for covering polyhedra as a proper generalization of the Fulkerson property, to which it reduces for F={0, 1} . We prove that the anti-join operation preserves the F -property. This implies the characterization of the coefficients of the facet-defining inequalities for the cycle and cocycle polyhedra associated with graphs noncontractible to the four-wheel W 4.