To explore the dynamic compression behavior and damage evolution characteristics of “rigid-flexible coupling” surrounding rock supporting structure. Split Hopkinson pressure bar (SHPB) was used to carry out constant amplitude cyclic bi-directional impact compression tests on the combination specimens of “rigid-rigid: rock (sandstone)-like material (RLM) - normal cement mortar (NCM)", “rigid-flexible: RLM - rubber cement mortar (RCM)" and “flexible-rigid: RCM - high-performance supporting concrete (HPSC)". The characteristics of stress-strain curve evolution, stress attenuation, strain growth, energy transformation, damage evolution characteristics, and damage state were compared and analyzed. The results show that compared with RLM-NCM, RLM-RCM and RCM-HPSC had remarkable dynamic yield ductility and excellent impact resistance. RCM and HPSC can work together to hinder the dynamic damage development of “rigid-flexible coupling” supporting structure and surrounding rock. Based on the one-dimensional stress wave propagation theory, the stress wave propagation models of rigid-flexible combination specimens under SHPB cyclic bi-directional impact were established, and the corresponding stress wave propagation equations were derived, which profoundly revealed the propagation law and attenuation mechanism of stress waves in SHPB impact rigid-flexible combination tests. Finally, based on the energy drive theory and residual energy theory, the impact instability energy criteria of surrounding rock under the rigid component combined support system and “rigid-flexible coupling” component combined support system were analyzed and discussed.