Using density functional theory within the generalized gradient approximation, we have theoretically studied the formation of neutral metal-aromatic complexes R-1-M and R-1-M-R-2, where M is either neutral lithium, calcium, or gallium and R-1 or R-2 is benzene or borazine. We first find that calcium atom is an effective mediator for cooperative formation of a sandwich complex with borazine, while others are not. When benzene and borazine are mixed in the presence of calcium, a 1:2:1 mixture of benzene-calcium-benzene, borazine-calcium-benzene, and borazine-calcium-borazine is expected. An "A"-shaped structure is predicted for homo- and heterocomplexes of borazine with partial B-B and B-C bonds, while two rings are planar in the case of homocomplexes of benzene. Our analysis of the electron density distributions in HOMO-1 to LUMO in terms of orbital symmetry in conjunction with analysis of l,m-projected electronic local density of states shows that this correlates with the charge transfer and the interaction of pi* states of the rings mediated by empty d-states of Ca, which is ultimately related to the polarity of the B-N bond. We find that there is a large accumulation of electron density on particular atoms upon complex formation, predicting characteristic behavior in electron-transfer reaction and nucleophilic reaction different from those for pure benzene or borazine molecule. The hetero-sandwich complex is of particular interest due to its asymmetrical distribution of excess electrons.