  Amplitude models are constructed to describe the resonance structure of
${D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}}$ and ${D^{0} \to
K^{+}\pi^{-}\pi^{-}\pi^{+}}$ 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 $D^{0} \to K_1(1270/1400)^{+}
\pi^{-}$ being prominent in ${D^{0}\to K^{-}\pi^{+}\pi^{+}\pi^{-}}$ and
$D^{0}\to K^{+}\pi^{-}\pi^{-}\pi^{+}$, 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 $R_{K3\pi} = 0.459\pm
0.010\,(\mathrm{stat}) \pm 0.012\,(\mathrm{syst}) \pm 0.020\,(\mathrm{model})$,
which is consistent with direct measurements. These models will be useful in
future measurements of the unitary-triangle angle $\gamma$ and studies of charm
mixing and $C\!P$ violation.
