jats:titleAjats:scbstract</jats:sc> </jats:title>jats:pIn this paper we present a detailed study of the four-body decay jats:italicB</jats:italic>jats:sup0</jats:sup> → jats:italicK</jats:italic>jats:sup+</jats:sup>jats:italicπ</jats:italic>jats:supjats:italic−</jats:italic></jats:sup>jats:italicℓ</jats:italic>jats:sup+</jats:sup>jats:italicℓ</jats:italic>jats:supjats:italic−</jats:italic></jats:sup>, where tensions with the Standard Model predictions have been observed. Our analysis of the decay with P- and S-wave contributions to the jats:italicK</jats:italic>jats:sup+</jats:sup>jats:italicπ</jats:italic>jats:supjats:italic−</jats:italic></jats:sup> 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 jats:italicB</jats:italic>jats:sup0</jats:sup> → jats:italicK</jats:italic>jats:sup+</jats:sup>jats:italicπ</jats:italic>jats:supjats:italic−</jats:italic></jats:sup>jats:italicℓ</jats:italic>jats:sup+</jats:sup>jats:italicℓ</jats:italic>jats:supjats:italic−</jats:italic></jats:sup> 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 jats:inline-formulajats:alternativesjats:tex-math$$ {m}{K{\pi }} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:msub mml:mim</mml:mi> mml:msub mml:miK</mml:mi> mml:miπ</mml:mi> </mml:msub> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula>. 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.</jats:p>jats:pThe 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 optimized 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 and the parametrization of the jats:italicB</jats:italic>jats:sup0</jats:sup> → jats:italicK</jats:italic>jats:sup+</jats:sup>jats:italicπ</jats:italic>jats:supjats:italic−</jats:italic></jats:sup>jats:italicℓ</jats:italic>jats:sup+</jats:sup>jats:italicℓ</jats:italic>jats:supjats:italic−</jats:italic></jats:sup> differential decay rate. 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.</jats:p>