A zone-folding construction is applied to the honeycomb lattice band structure to yield explicit expressions for the Huckel pi-molecular orbitals, energies and symmetries of trivalent polyhedra consisting of hexagons and squares [(4,6) cages] with octahedral symmetry. The A(1), A(2), and E representations are accessible in this way, but not the T-1 and T-2 representations. Therefore, we have also performed numerical Hockel calculations on a large set of cages. A clear distinction in electronic structure between leapfrog, nonleapfrog type 1 and nonleapfrog type 2 cages is revealed. The results are relevant both for carbon cages and alternating boron-nitride cages. Quantum chemical calculations on C-24, C-56, C-72, and B36N36 confirm the results.