Nonlinear oscillators that are mutually coupled via dissimilar (or conjugate) variables display distinct regimes of synchronous behavior. In identical chaotic oscillators diffusively coupled in this manner, complete synchronization occurs only by chaos suppression when the coupled subsystems drive each other into a regime of periodic dynamics. Furthermore, the coupling does not vanish but acts as an "internal" drive. When the oscillators are mismatched, phase synchronization occurs, while in a master slave configuration, generalized synchrony results. These effects are demonstrated in a system of coupled chaotic Rossler oscillators.