Dyons are objects with both electric and magnetic charges. I will present a one-parameter family of stationary, asymptotically at analytical solutions of the Einstein-Maxwell equations with only a mild singularity, which are endowed with mass, angular momentum, a dipole magnetic moment and a quadrupole electric moment. I will discuss the various clues which point out to a complex underlying physical system. This field configuration is generated by two co-rotating extreme black holes, which are both electromagnetic and gravitational dyons, and are held apart by an electrically charged, magnetised rod which also acts as a Dirac-Misner string.