Endohedral and exohedral polyhedral cage molecules of the form (HAO(3/2))(8) (A = C, Si, Ge) with double four-membered ring D4R units complexed with the atomic or ionic species (Li+, Na+, K+, F-, Cl-, Br-, He, Ne, Ar) have been investigated at the B3LYP/6-31G(d) and B3LYP/6-311++G(d,p) levels. Geometric, electronic, and energetic properties were obtained. For the endohedral complexes the noble gas atoms (X = He, Ne, and Ar) inside the cage cause the cages to expand, and the extent of the expansion depends on the size of the included atom. Endohedral alkali ions, in contrast, exhibit both attractive and repulsive interactions with the cage atoms. The cage expands when X = K+ and contracts when X = Li+ or Na+ for A = Si, Ge, and for A = C, the cage expands for all three ions. Encapsulation of the halide ions results in cage expansion throughout. Furthermore, the symmetry of the endohedral complexes when X is a cation depends critically on the relative cation and cage sizes. The binding energies of the endohedral and exohedral complexes document a clear preference for the latter, except for halides, where the endohedral complexes are more stable. The stability of endohedral complexes containing the isoelectronic species X = Na+, Ne, F- is determined by the charge transfer to the A-O cage bonding sites. The formation of the endohedral complexes is discussed in terms of transition states that connect the exohedral and endohedral minima as well as the activation barriers for insertion of the guest into the cage. Our studies predict that a fluoride anion can penetrate into the (HAO(3/2))(9) cage without destroying it. For X = Cl-, in contrast, the cage ruptures upon insertion of the impurity.