The unusual stability of the experimentally known 1,3-dehydro-5,7-adamantyl dication was previously explained by four-center two-electron aromaticity with three-dimensional (tetrahedral) topology. Magnetic criteria (ACID and ring-current analysis) now demonstrate that there is also a very strong contribution from hyperconjugation with all six methylene bridges. The delocalized system of electrons thus includes all valence electrons, and the structure, therefore, should rather be described as a spherically aromatic 50-electron system. The corresponding T-d-symmetric tetracation with 48 electrons is antiaromatic and not a minimum structure. With a He atom or a hydride ion at the center of the cage, the tetracation is predicted to form a kinetically stable complex. Magnetic criteria demonstrate that the antiaromaticity is greatly reduced, and a bond analysis hints at bonding interactions between He (and H-) and the carbon atoms of the adamantane cage.