The electronic structures of conjugated polymers containing methyl radicals, carbenes, and nitrogen-based radicals coupled in various ways through benzene rings are examined, employing band-structure calculations in the extended Huckel approximation. The structural and electronic properties of polymers with a para-phenylene or meta-phenylene coupling unit are compared. In the polymer with methyl radicals coupled through a para-phenylene unit, a pairing or Peierls distortion occurs to remove the degeneracy at the Fermi level. The resulting bandgap is nevertheless relatively small; we conclude that such polymers are likely to exhibit high electrical conductivity upon doping, very much like polyacetylene. On the other hand, in the polymers with a meta-phenylene coupling unit, striking symmetry-determined, half-occupied narrow bands appear at the Fermi level and contribute to the stability of the ferromagnetic state. The relation of a potential ferromagnetic state to metallic, CDW, and SDW states is discussed from the viewpoint of orbital interactions in extended systems. We suggest novel 3- and 4-fold helical structures for the metaphenylene-coupled polymers.