We provide a detailed study of hydrogen bonding arrangements, relative stability, residual entropy, and an analysis of the many-body effects in the (H2O)20 (D-cage), (H2O)24 (T-cage), and (H2O)28 (H-cage) hollow cages making up structures I (sI) and II (sII) of clathrate hydrate lattices. Based on the enumeration of the possible hydrogen bonding networks for a fixed oxygen atom scaffold, the residual entropy (S0) of these three gas phase cages was estimated at 0.754 82, 0.754 44, and 0.754 17 · Nkb, where N is the number of molecules and kb is Boltzmann's constant. A previously identified descriptor of enhanced stability based on the relative arrangement and connectivity of nearest-neighbor fragments on the polyhedral water cluster [strong-weak-effective-bond model] also applies to the larger hollow cages. The three cages contain a maximum of 7, 9, and 11 such preferable arrangements of trans nearest dimer pairs with one "free" OH bond on the donor molecule (t1d dimers). The Many-Body Expansion (MBE) up to the 4-body suggests that the many-body terms vary nearly linearly with the cluster binding energy. Using a hierarchical approach of screening the relative stability of networks starting from optimizations with the TIP4P, TTM2.1-F, and MB-pol classical potentials, subsequently refining at more accurate levels of electronic structure theory (DFT and MP2), and finally correcting for zero-point energy, we were able to identify a group of four low-lying isomers of the (H2O)24 T-cage, two of which are antisymmetric and the other two form a pair of antipode configurations.