Amplitude models are constructed to describe the resonance structure of ${D^{0}\to K^{-}π^{+}π^{+}π^{-}}$ and ${D^{0} \to K^{+}π^{-}π^{-}π^{+}}$ 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 $3.0\mathrm{fb}^{-1}$. The largest contributions to both decay amplitudes are found to come from axial resonances, with decay modes $D^{0} \to a_1(1260)^{+} K^{-}$ and $D^{0} \to K_1(1270/1400)^{+} π^{-}$ being prominent in ${D^{0}\to K^{-}π^{+}π^{+}π^{-}}$ and $D^{0}\to K^{+}π^{-}π^{-}π^{+}$, respectively. Precise measurements of the lineshape parameters and couplings of the $a_1(1260)^{+}$, $K_1(1270)^{-}$ 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 $R_{K3π} = 0.459\pm 0.010\,(\mathrm{stat}) \pm 0.012\,(\mathrm{syst}) \pm 0.020\,(\mathrm{model})$, 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 $C\!P$ violation.