Amplitude models are constructed to describe the resonance structure of D0→ K-π+π+π- and D0→ K+π-π-π+ decays using pp collision data collected at centre-of-mass energies of 7 and 8 TeV with the LHCb experiment, corresponding to an integrated luminosity of 3.0 fb- 1. The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes D0→ a1(1260) +K- and D0→ K1(1270 / 1400) +π- being prominent in D0→ K-π+π+π- and D0→ K+π-π-π+, respectively. Precise measurements of the lineshape parameters and couplings of the a1(1260) +, K1(1270) - and K(1460) - resonances are made, and a quasi model-independent study of the K(1460) - resonance is performed. The coherence factor of the decays is calculated from the amplitude models to be RK3π=0.459±0.010(stat)±0.012(syst)±0.020(model), which is consistent with direct measurements. These models will be useful in future measurements of the unitary-triangle angle γ and studies of charm mixing and CP violation.