Designing porous materials which can selectively adsorb CO2 or CH4 is an important environmental and industrial goal which requires an understanding of the host-guest interactions involved at the atomic scale. Metal-organic polyhedra (MOPs) showing permanent porosity upon desolvation are rarely observed. We report a family of MOPs (Cu-1a, Cu-1b, Cu-2), which derive their permanent porosity from cavities between packed cages rather than from within the polyhedra. Thus, for Cu-1a, the void fraction outside the cages totals 56% with only 2% within. The relative stabilities of these MOP structures are rationalized by considering their weak nondirectional packing interactions using Hirshfeld surface analyses. The exceptional stability of Cu-1a enables a detailed structural investigation into the adsorption of CO2 and CH4 using in situ X-ray and neutron diffraction, coupled with DFT calculations. The primary binding sites for adsorbed CO2 and CH4 in Cu-1a are found to be the open metal sites and pockets defined by the faces of phenyl rings. More importantly, the structural analysis of a hydrated sample of Cu-1a reveals a strong hydrogen bond between the adsorbed CO2 molecule and the Cu(II)-bound water molecule, shedding light on previous empirical and theoretical observations that partial hydration of metal-organic framework (MOF) materials containing open metal sites increases their uptake of CO2. The results of the crystallographic study on MOP-gas binding have been rationalized using DFT calculations, yielding individual binding energies for the various pore environments of Cu-1a.