This paper describes the geometrical and electronic structure of a hypothetical 3,4-connected tetragonal allotrope of carbon (space group P42/mmc, No. 131) built from 1,4-cyclohexadiene rings. The three-dimensional net, containing trigonal and tetrahedral atoms in a ratio of 2:1, has a calculated density of 3.12 g/cm3, intermediate between graphite and diamond. Band structure calculations for this net have been performed using the extended Hückel method. One-dimensional substructures of a polyquinoid, polyspiroquinoid, and polycyclophane nature are instructive in approaching the electronic structure of the full net. These substructures point to the importance of through-space interactions of the stacked olefin units in the net, separated by only 2.53 Å. It is apparent that interaction leads to substantial dispersion of the π and π* bands, the highest occupied and lowest unoccupied bands in the tetragonal structure, respectively. The resulting density of states profile is that of a metal, with a significant π-π* band overlap at the Fermi level. Related nets formed by substituting boron and nitrogen into the trigonal sites of the lattice are studied as well. The electron count on the atom in the trigonal sites in the lattice significantly affects the band structure about the Fermi level; B2C should be metallic, and CN2 an insulator.