The four chemically significant spherical manifolds of atomic orbitals are the sp(3), sd(5), sp(3)d(5), and sd(5)f(7) manifolds of 4, 6, 9, and 13 orbitals found in the chemistry of the main-group elements, the early transition metal homoleptic hydrides and alkyls, most other compounds of the d-block transition metals, and the actinides; respectively. Coordination geometries with an inversion center (e.g., the octahedron) or a unique reflection plane passing through no vertices (e.g., the trigonal prism) are symmetry forbidden for the sd5 manifold thereby accounting for some unusual experimental and computed geometries for six-coordinate early transition metal hydrides and alkyls. The maximum coordination numbers for polyhedra with inversion centers for the nine-orbital sp(3)d(5),d 13-orbital sd(5)f(7) manifolds are 6 and 12 corresponding to the regular octahedron and regular icosahedron, respectively, for the most symmetrical manifestations of these coordination numbers.