  An amplitude analysis of ${B^0\to (\pi^+\pi^-) (K^+\pi^-)}$ decays is
performed in the two-body invariant mass regions ${300 < m(\pi^+\pi^-)<1100}$
MeV/$c^2$, accounting for the $\rho^0$, $\omega$, $f_0(500)$, $f_0(980)$ and
$f_0(1370)$ resonances, and ${750 < m(K^+\pi^-)<1200}$ MeV/$c^2$, which is
dominated by the $K^{*}(892)^0$ meson. The analysis uses 3 fb$^{-1}$ of
proton-proton collision data collected by the LHCb experiment at centre-of-mass
energies of 7 and 8 TeV. The $CP$ averages and asymmetries are measured for the
magnitudes and phase differences of the contributing amplitudes. The
$CP$-averaged longitudinal polarisation fractions of the vector-vector modes
are found to be $\tilde{f}^0_{\rho K^{*}} = 0.164 \pm 0.015 \pm 0.022$ and
$\tilde{f}^0_{\omega K^{*}} = 0.68 \pm 0.17 \pm 0.16$, and their $CP$
asymmetries, $\mathcal{A}^0_{\rho K^{*}} = -0.62 \pm 0.09 \pm 0.09$ and
$\mathcal{A}^0_{\omega K^{*}} = -0.13 \pm 0.27 \pm 0.13$, where the first
uncertainty is statistical and the second systematic.
