In this paper we systematically investigate strong interactions between two like-signed quasi-geostrophic vortices containing different uniform potential vorticity. The interaction depends on six parameters: the potential vorticity ratio between the two vortices, their volume ratio, their individual height-to-width aspect ratio, their vertical offset, and their horizontal separation distance. We first determine the conditions under which a strong interaction may occur. To that end, we calculate equilibrium states using an asymptotic approach which models the vortices as ellipsoids and we additionally assess their linear stability. It is found that vortices having similar potential vorticity interact strongly (e.g. merge) at closer separation distances than do vortices with a dissimilar potential vorticity. This implies that interactions between vortices having significantly different potential vorticity may be more destructive, for a given separation distance. This is confirmed by investigating the nonlinear evolution of the vortices over a subset of the full parameter space, solving the full dynamical quasi-geostrophic equations. Many forms of interaction occur, but merger or partial merger (where the largest vortex grows in volume) is mostly observed for interactions between vortices of similar potential vorticity.