One of the most exciting recent (2009) discoveries in metal cluster chemistry is the pentagonal prismatic Co@Ge-10(3-) ion, found in [K(2,2,2-crypt)](4)[Co@Ge-10][Co(1,5-C8H12)(2)]center dot toluene and characterized structurally by X-ray diffraction. The complete absence of triangular faces in the pentagonal prismatic structure of Co@Ge-10(3-) contradicts expectations from the well-established Wade-Mingos rules, which predict polyhedral structures having mainly or entirely triangular faces. A theoretical study on Co@Ge-10(z) systems (z = -5 to +1) predicts a singlet D-Sh pentagonal prismatic global minimum for the trianion Co@Ge-10(3-) in accord with this experimental result. Redox reactions on this pentagonal prismatic Co@Ge-10(3-) trianion generate low-energy pentagonal prismatic structures for Co@Ge-10(z) where z = 0, -1, -2, -4, and -5 having quartet, triplet, doublet, doublet, and triplet spin states, respectively. Similar theoretical methods predict a singlet C-3v polyhedral structure for the monoanion Co@Ge-10(-), similar to previous theoretical predictions on the isoelectronic neutral Ni@Ge-10 and the structure realized experimentally in the isoelectronic Ni@In-10(10-) found in the K10In10Ni intermetallic. Redox reactions on this C-3v polyhedral Co@Ge-10(-) monoanion generate low energy C-3v polyhedral structures for Co@Ge-10(z) where z = 0, -2, -3, and -4 having doublet, doublet, triplet, and quartet spin states, respectively.