  In this paper we present a detailed study of the four-body decay $B^0\to
K^{+}\pi^{-}\ell^{+}\ell^{-}$, where tensions with the Standard Model
predictions have been observed. Our analysis of the decay with P- and S-wave
contributions to the $K^{+}\pi^{-}$ system develops a complete understanding of
the symmetries of the distribution, in the case of massless and massive
leptons. In both cases, the symmetries determine relations between the
observables in the $B^0\to K^{+}\pi^{-}\ell^{+}\ell^{-}$ decay distribution.
This enables us to define the complete set of observables accessible to
experiments, including several that have not previously been identified. The
new observables arise when the decay rate is written differentially with
respect to $m_{K\pi}$. We demonstrate that experiments will be able to fit this
full decay distribution with currently available data sets and investigate the
sensitivity to new physics scenarios given the experimental precision that is
expected in the future. The symmetry relations provide a unique handle to
explore the behaviour of S-wave observables by expressing them in terms of
P-wave observables, therefore minimising the dependence on poorly-known S-wave
form factors. Using this approach, we construct two theoretically clean S-wave
observables and explore their sensitivity to new physics. By further exploiting
the symmetry relations, we obtain the first bounds on the S-wave observables
using two different methods and highlight how these relations may be used as
cross-checks of the experimental methodology. We identify a zero-crossing point
that would be at a common dilepton invariant mass for a subset of P- and S-wave
observables, and explore the information on new physics and hadronic effects
that this zero point can provide.
