Horizon exchange and the universal all-spin response of Kerr

The radial Teukolsky equation is invariant under exchange of the outer and inner horizons, although the physical boundary condition is not. We derive an exact monodromy relation between the corresponding Mano--Suzuki--Takasugi (MST) near-zone responses. At leading order in frequency and to all orders in spin, this relation yields, for every multipole, an expression for the leading horizon-exchange-odd response. For gravitational perturbations, it determines the exchange-odd contribution to quadrupolar absorption and organizes the exchange-even dynamical response into a digamma function whose expansion accounts for the odd- and even-zeta structures found in Kerr-Compton calculations. The exact relation, rather than the individual previously known small-spin coefficients, is the central result.