Abstract Amplitude models are constructed to describe the resonance structure of $${D^{0}\rightarrow K^{-}\pi ^{+}\pi ^{+}\pi ^{-}}$$ D0→K-π+π+π- and $${D^{0} \rightarrow K^{+}\pi ^{-}\pi ^{-}\pi ^{+}}$$ 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 $$fb^{-1}$$ fb-1 . The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes $$D^{0} \rightarrow a_1(1260)^{+} K^{-}$$ D0→a1(1260)+K- and $$D^{0} \rightarrow K_1(1270/1400)^{+} \pi ^{-}$$ D0→K1(1270/1400)+π- being prominent in $${D^{0}\rightarrow K^{-}\pi ^{+}\pi ^{+}\pi ^{-}}$$ D0→K-π+π+π- and $$D^{0}\rightarrow K^{+}\pi ^{-}\pi ^{-}\pi ^{+}$$ D0→K+π-π-π+ , respectively. Precise measurements of the lineshape parameters and couplings of the $$a_1(1260)^{+}$$ a1(1260)+ , $$K_1(1270)^{-}$$ K1(1270)- and $$K(1460)^{-}$$ K(1460)- resonances are made, and a quasi model-independent study of the $$K(1460)^{-}$$ K(1460)- resonance is performed. The coherence factor of the decays is calculated from the amplitude models to be $$R_{K3\pi } = 0.459\pm 0.010\,(\mathrm {stat}) \pm 0.012\,(\mathrm {syst}) \pm 0.020\,(\mathrm {model})$$ 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 $$\gamma $$ γ and studies of charm mixing and $$C\!P$$ CP violation.