We analyze the many-body exchange interactions in atomic and molecular clusters as they arise in the supermolecular SCF and MP2 approaches. A rigorous formal setting is provided by the symmetry-adapted perturbation theory. Particular emphasis is put on the decomposition into the single exchange (SE) and triple exchange (TE) terms, at the SCF and correlated levels. We also propose a novel approach, whereby selected SE nonadditive exchange terms are evaluated indirectly, as differences of the two-body SAPT corrections arising between the components of the trimer treated as a complex of a dimer and a monomer (pseudodimer approach). This provides additional insights into the nature of various nonadditive effects, an interpretation of supermolecular interaction energies, and may serve as a viable alternative for the calculation of some SE terms.