We find self-accelerating wavepackets that are exact solutions of the Dirac equation, for massive and massless fermions and bosons. These Dirac particles mimic the dynamics of a free-charge moving under a “virtual” electric and/or magnetic field, although no field is acting and there is no charge. In all cases, the solutions exhibit shape-preserving acceleration/rotation, and appear exactly the same for observers in different Lorentz frames. We show that the dynamics of such a self-accelerating solution displays the "twin-paradox", even though no force causes it to accelerate.