We present an illustrative discussion of the physics potential of μ⁻–e⁺ conversion. We point out that this process, although much less frequently studied than the related but much more popular processes of μ⁻–e⁻ conversion and neutrinoless double beta decay, may in fact be a promising alternative to detect both lepton flavour and number violation. However, for this goal to be reached, a combined effort of experiments and theory, both in nuclear and particle physics, is necessary to advance. The aim of this paper is to be an “appetiser” to trigger such an initiative.