There is a strong interest in the collective behavior of chiral active particles that can propel and rotate themselves. In the presence of alignment interactions, chiral self-propulsion can induce vortex patterns in the velocity fields which are non-permanent and do not induce global vorticity. Here we combine theoretical arguments and computer simulations to predict a so-far unknown class of collective behavior. We show that, for chiral active particles, vortices with significant dynamical coherence emerge spontaneously. They originate from the interplay between attraction interactions and chirality in the absence of alignment interactions. Depending on parameters, the vortices can either feature a constant vorticity or a vorticity that oscillates periodically in time, resulting in self-reverting vortices. Our results may guide future experiments to realize customized collective phenomena such as spontaneously rotating gears and patterns with a self-reverting order.