In computer simulations, long-range electrostatic interactions are surprisingly well approximated by truncated, short-ranged pair potentials. Examples are reaction field methods; the Wolf method; and a number of schemes based on cancellation of electric multipole moments inside a cut-off region. These methods are based on the assumption that the polarization of the neglected surroundings can be inferred from a local charge distribution. Multipole moments themselves are only approximations to the true charge distribution, approximations which many times are needed to simplify calculations in complex systems. In this work we investigate a new, generalized pair-potential based on the idea of moment cancellation that covers interactions between electrostatic moments of any type. We find that moment cancellation in itself is insufficient to generate accurate results and a more restricted formalism is needed, in our case to cancel the virtual charges of the imposed moments. Thus, it is unfeasible to cancel higher-order moments with explicit higher-order moments such as dipoles and instead image charges are needed. The proposed pair-potential is general and straight forwardly implementable for any electrostatic moment – monopole, dipole, quadrupole, etc. – with a computational complexity scaling with the number of particles in the system.