We report three surprising results regarding the nature of the spatial broken symmetries in the two-dimensional (2D), quarter-tilled band with strong electron-electron interactions that provides a microscopic model of the 2:1 cationic organic charge-transfer solids (CTS's). First, in direct contradiction to the predictions of one-electron theory, we find a coexisting "bond-order and charge-density wave" (BCDW) insulating sound state in the 2D rectangular lattice for all anisotropies, including the isotropic limit. Second, in contrast to the interacting half-filled band, which exhibits one singlet-to-antiferromagnet (AFM) transition as the interchain coupling is increased from zero, there occur in the interacting quarter-filled band two distinct transitions: a similar singlet-to-antiferromagnet/spin-density wave (AFM/SDW) transition at small interchain coupling, giving rise to a bond-charge-spin density wave (BCSDW) state, followed by a second AFM/SDW-to-singlet transition at large interchain coupling. Third, we show that our conclusions remain unchanged if one assumes the conventional "effective 1/2-filled" lattice of dimer sites for the CTS's: the dimer lattice unconditionally dimerizes again to give the same BCDW found in the quarter-filled band. We make detailed comparisons to recent experiments in the tetramethyl-tetrathiafulvalene (TMTTF), tetramethyl-tetraselenafulvalene (TMTSF), bisethylenedithio-tetrathiafulvalene (BEDT-TTF) and bisethylenedithio-tetraselenafulvalene (BETS)-based CTS's. Our theory explains the mixed charge-spin density waves observed in TMTSF and certain BEDT-TTF systems, as well as the absence of antiferromagnetism in the BETS-based systems. An important consequence of this work is the suggestion that organic superconductivity is related to the proximate Coulomb-induced BCDW, with the SDW that coexists for large anisotropies being also a consequence of the BCDW, rather than the driver of superconductivity. We point out that the BCDW and BCSDW states are analogous to four different classes of "paired'' semiconductors that are obtained within certain models of exotic superconductivity. That all four of these models can in principle give rise to superconductivity in the weakly incommensurate regime provides further motivation for the notion that the BCDW may be driving the superconductivity in the organics.
