The resonant substructure of D^0 →π^+π^-π^+π^- decays is studied using data collected by the CLEO-c detector. An amplitude analysis is performed in order to disentangle the various intermediate state contributions. To limit the model complexity a data driven regularization procedure is applied. The prominent contributions are the decay modes D^0→ a_1(1260)^+π^-, D^0→σ f_0(1370) and D^0 →ρ(770)^0 ρ(770)^0. The broad resonances a_1(1260)^+, π(1300)^+ and a_1(1640)^+ are studied in detail,

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... g quasi-model-independent parametrizations of their lineshapes. The mass and width of the a_1(1260)^+ meson are determined to be m_a_1(1260)^+=[1225±9(stat)±17(syst)±10(model)]MeV/c^2 and Γ_a_1(1260)^+=[430±24(stat)±25 (syst)±18(model)]MeV. The amplitude model of D^0→ K^+K^-π^+π^- decays obtained from CLEO II.V, CLEO III, and CLEO-c data is revisited with improved lineshape parametrizations. The largest components are the decay modes D^0→ϕ(1020)ρ(770)^0, D^0 → K_1(1270)^+ K^- and D^0 → K(1400)^+ K^-. The fractional CP-even content of the decay D^0→π^+π^-π^+π^- is calculated from the amplitude model to be F_+^4π=[72.9±0.9(stat)±1.5(syst)±1.0(model)]%, consistent with that obtained from a previous model-independent measurement. For D^0 → K^+K^-π^+π^- decays, the CP-even fraction is measured for the first time and found to be F_+^KKππ=[75.3±1.8(stat)±3.3(syst)±3.5(model)]%. The global decay rate asymmetries between D^0 and D^0 decays are measured to be A^4π_CP=[+0.54±1.04(stat)±0.51(syst)]% and A^KKππ_CP=[+1.84±1.74(stat)±0.30(syst)]%.