The conventional (excluding non-Kekulé, singlet diradical structures) quinones of benzocyclobutadiene were studied computationally. Eight structures were examined, namely (based on the CA names for benzocyclobutenedione), benzocyclobutenedione or bicyclo[4.2.0]octa-1,3,5-triene-7,8-dione, bicyclo[4.2.0]octa-3,5,8-triene-2,7-dione, bicyclo[4.2.0]octa-1,4,6-triene-3,8-dione, bicyclo[4.2.0]octa-1(6),4,7-triene-2,3-dione, bicyclo[4.2.0]octa-1(8), 4,6-triene-2,3-dione, bicyclo[4.2.0]octa-1(6),3,7-triene-2,5-dione, bicyclo[4.2.0]octa-1(8),3,6-triene-2,5-dione, and bicyclo[4.2.0]octa-1,5,7-triene-3,4-dione (the question of resonance or tautomerism for the 2,3-dione pair and the 2,5-dione pair is considered). Using DFT (B3LYP/6-31G*) and ab initio (MP2/6-31G*) methods the geometries of the eight species were optimized, giving similar results for the two methods. The heats of formation of the quinones were calculated, placing them in low-energy (-17 kJ mol(-1), 7,8-dione), medium-energy (79-137 kJ mol(-1), 2,7-, 3,8-, and 3,4-diones), and high-energy (260-275 kJ mol(-1), 2,3- and 2,5-diones) groups. Diels-Alder reactivity as dienophiles with butadiene indicated the 2,7-, 3,8-, and particularly the 3,4-quinone may be relatively unreactive toward dimerization or polymerization and are attractive synthesis goals. Isodesmic ring-opening reactions and NICS calculations showed aromatic/nonaromatic properties to be essentially as expected from the presence of a benzene or cyclobutadiene ring. UV spectra, ionization energy electron affinity, and HOMO/LUMO energies were also calculated.