  We establish a new relation between classical observables for scattering and
bound orbits of a massive probe particle in a Kerr background. We find an exact
representation of the Hamilton-Jacobi action in terms of the conserved charges
which admits an analytic continuation, both for the radial and polar
contribution, for a general class of geodesics beyond the equatorial case.
Remarkably, this allows to extend the boundary to bound dictionary and it
provides an efficient method to compute the deflection angles and the time
delay for scattering orbits, as well as frequency ratios for bound orbits, in
the probe limit but at all orders in the perturbative expansion.
