The excitation curve of e+e- annihilation into four charged pions in the rho(770) region is calculated using three existing models with rho mesons and pions in intermediate states supplemented by Feynman diagrams with the a1(1260)pi intermediate states. A two-term phenomenological Lagrangian of the a1rhopi interaction is used. The mixing angle is determined by fitting the e+e--->pi+pi-pi+pi- cross section data of the Novosibirsk CMD-2 Collaboration and also its combination with the low-energy part of the BABAR Collaboration data. It is shown that the inclusion of the a1pi intermediate states succeeds in obtaining a good agreement with the data on both cross section and the rho0-->pi+pi-pi+pi- decay width. When moving to energies above 1 GeV, the rho(1450) and rho(1700) resonances are taken into account to get excellent agreement with the BABAR data over the full energy range up to 4.5 GeV.