We investigate soliton excitations in halogen (X)-bridged binuclear transition-metal (M) complexes (MMX chains), which are essentially characterized by competing intrasite and intersite electron-lattice couplings. Employing a simple electron-phonon model, their spatial and energy structures are calculated for two distinct, ground states of MMX chains. In the weak-coupling region. all the solitons are degenerate and have close analogy with those in trans-polyacetylene.