  We present two exact relations, valid for any dilepton invariant mass region
(large and low-recoil) and independent of any effective Hamiltonian
computation, between the observables $P_i$ and $P_i^{CP}$ of the angular
distribution of the 4-body decay $B \to K^*(\to K\pi) l^+l^-$. These relations
emerge out of the symmetries of the angular distribution. We discuss the
implications of these relations under the (testable) hypotheses of no scalar or
tensor contributions and no New Physics weak phases in the Wilson coefficients.
Under these hypotheses there is a direct relation among the observables
$P_{1}$,$P_2$ and $P_{4,5}^\prime$. This can be used as an independent
consistency test of the measurements of the angular observables. Alternatively,
these relations can be applied directly in the fit to data, reducing the number
of free parameters in the fit. This opens up the possibility to perform a full
angular fit of the observables with existing datasets. An important consequence
of the found relations is that a priori two different measurements, namely the
measured position of the zero ($q_0^2$) of the forward-backward asymmetry
$A_{FB}$ and the value of $P_5^\prime$ evaluated at this same point, are
related by $P_4^2(q_0^{2})+P_5^2(q_0^{2})=1$. Under the hypotheses of real
Wilson coefficients and $P_4^\prime$ being SM-like, we show that the higher the
position of $q_0^{2}$ the smaller should be the value of $P_5^\prime$ evaluated
at the same point. A precise determination of the position of the zero of
$A_{FB}$ together with a measurement of $P_4^\prime$ (and $P_1$) at this
position can be used as an independent experimental test of the anomaly in
$P_5^\prime$. We also point out the existence of upper and lower bounds for
$P_1$, namely $P_5^{\prime 2}-1 \leq P_1 \leq 1-P_4^{\prime 2}$, which
constraints the physical region of the observables.
