This paper addresses the coordination challenge at intersections of mixed traffic involving both Human-Driven Vehicles (HDVs) and Connected and Autonomous Vehicles (CAVs). To strike a balance between coordination performance and safety guarantees, we propose an invariant safe Contingency Model Predictive Control (CMPC) framework. The CMPC framework incorporates two parallel horizons for the ego vehicle: a nominal horizon optimized for performance based on the most likely prediction of the opponent HDV, and a contingency horizon designed to maintain an invariant safe backup plan for emergencies. In the contingency horizon, we consider the worst-case behavior of the human driver and formulate safety constraints using the forward reachable sets of the HDV within the planning horizon. These safety constraints are complemented by maximal invariant safe sets as terminal constraints. The two horizons are tied together by enforcing equality of the feedback inputs at the beginning of the horizons. We provide theoretical evidence supporting the recursive feasibility and persistent performance improvement of the invariant safe CMPC compared to our previously proposed nominal invariant safe Model Predictive Control (MPC). Through simulation studies, we evaluate the proposed method. The simulation results demonstrate that the CMPC approach achieves enhanced performance by reducing conservatism while simultaneously preserving the invariant safety property.