  An ion and a polar molecule interact by an anisotropic ion-dipole potential
scaling as $- \alpha \cos (\theta)/r^2$ at large distances. Due to its
long-range character, it modifies the properties of angular wave functions,
which are no longer given by spherical harmonics. In addition, an effective
centrifugal potential in the radial equation can become attractive for low
angular momenta. In this paper, we develop a general framework for an
ion-dipole reactive scattering, focusing on the regime of large $\alpha$. We
introduce modified spherical harmonics as solutions of the angular part of the
Schr\"odinger equation and derive several useful approximations in the limit of
large $\alpha$. We present a formula for the scattering amplitude expressed in
terms of the modified spherical harmonics and we derive expressions for the
elastic and reactive collision rates. The solutions of the radial equation are
given by Bessel functions, and we analyse their behaviour in two distinct
regimes corresponding, basically, to attractive and repulsive long-range
centrifugal potentials. Finally, we study reactive collisions in the universal
regime, where the short-range probability of loss or reaction is equal to
unity.
