The truncated spectral resolution of the exact exchange operator for closed-shells systems, first proposed by Andzelm et al. [J. Chem. Phys. 83, 4565 (1985)], and currently used in atomic, molecular, and solid-state ab initio calculations is revisited. We show how stating the exchange matrix elements in terms of projected functions over the subspace subtended by the exact exchange operator provides a means to extract valuable information about the quality of the approximation. Special interest is paid on two-center exchange interactions with truncated monocentric spectral resolution (TMSR) of the operator. We have undertaken numerical studies of the exact and TMSR exchange for the Li+–Li+, Li+–Cl−, Cl−–Li+, Cl−–Cl−, and He–He closed-shell pairs with an special attention to the fictitious asymmetry introduced by the approximation. Our main conclusions are: (a) the TMSR approximation for two-center exchange interactions deviates monotonically from the exact exchange with increasing inter