In aluminosilicate cage compounds Al-O-Al linkages are usually not found, in accord with the "Al avoidance" rule. In some cases Al-O-Al linkages can be stabilized by forming additional bonds to the bridging O atom. This can occur through direct coordination of cations or by the fusing of aluminosilicate rings to create "drum"-like molecules. We have calculated the structures, energetics, and NMR and vibrational spectra for several such aluminosilicate drum-like molecules and for related alumoxanes, as well as for analogous silicate molecules. Calculations on Si2Al4O6H8, Al6O6H6, Si4Al4O8H12, Si6O9H6, Si8O12H8, Si4Al4O12H84-, and Si4Al4O12H8Na4 reproduce the experimental structures (where available) and the observed trends in Si NMR shieldings. In the double 3-ring and double 4-ring (D3R and D4R) silicate cages Si6O9H6 and Si8O12H8 the Si atoms are shielded compared to their monomeric units, with a significantly greater shielding for the D4R compared to the D3R. The Si-4,Al-4 D4R bare anion has about the same Si shielding as Si8O12H8, while the neutral Si4Al4O12H8Na4 shows a 13 ppm deshielding. These trends in Si NMR shieldings are in accord with those observed in both the molecules and in aluminosilicate minerals. On the other hand, the fused or "drum"-like D3R and D4R rings show Si atoms and Al atoms which are calculated to be deshielded with respect to their corresponding monomers, in accord with experiment. We also reproduce the difference in frequency of the most intense IR absorptions for the various double-ring molecules. While silicate D3R cages are considerably more stable than corresponding D3R, the stabilities of the D3R molecule Al6O6H6 and the hypothetical D4R symmetry Al8O8H8 are quite similar. Calculated gas-phase proton affinities for Si6O9H6, Si2Al4O6H8, and Al6O6H6 are similar, although protonation causes cage breaking for Si2Al4O6H8 and Al6O6H6. For a drum-like D4R of composition Si4Al4O8H12, which is calculated to be stable with respect to Si2Al4O6H8 and Si4O4H8, we present a calculated structure as well as NMR properties.