Sharp phase transitions allow for the existence of a third family of stable compact stars, twin stars. In this work, we investigate for the first time the role of strong magnetic fields on non-magnetic twin-star sequences and the case in which magnetic fields themselves give rise to a third family of stable stars. We use three sets of equations of state to study such effects from a general point of view. Magnetic field effects are introduced in the structure of stars through the solution of the Einstein-Maxwell equations, assuming a poloidal magnetic field configuration and a metric that allows for the description of deformed stars. We show that strong magnetic fields can destabilize twin-star sequences. On the other hand, magnetic fields can also give rise to twin stars in models that did not predict these sequences. In this sense, magnetic fields can play an important role in the evolution of neutron stars.