  We study the coherent nonlinear tunneling dynamics of a binary mixture of
Bose-Einstein condensates in a double-well potential. We demonstrate the
existence of a new type of mode associated with the "swapping" of the two
species in the two wells of the potential. In contrast to the symmetry breaking
macroscopic quantum self-trapping (MQST) solutions, the swapping modes
correspond to the tunneling dynamics that preserves the symmetry of the double
well potential. As a consequence of two distinct types of broken symmetry MQST
phases where the two species localize in the different potential welils or
coexist in the same well, the corresponding symmetry restoring swapping modes
result in dynamics where the the two species either avoid or chase each other.
In view of the possibility to control the interaction between the species, the
binary mixture offers a very robust system to observe these novel effects as
well as the phenomena of Josephson oscillations and pi-modes
