We put forward a hybrid optimal control approach for joint optimization of cooperative (automated) vehicle trajectories and traffic signals for an intersection configured with different turning movements on multiple arms. The ride comfort, travel time, and throughput involving all vehicles are optimized by continuous vehicle acceleration and discrete traffic signal switch decisions subject to vehicle motion constraints on following gaps, speeds, accelerations, and upper bounds on the maximal signal stage lengths. The red time is designed as a concise vehicle position constraint to enable simultaneous evaluation of traffic-level and vehicle-level decisions. To decrease the computational burden of the mixed integer nonlinear program, the joint control formulation of an intersection is linearized and then decomposed using the Benders decomposition algorithm, generating a sequence of independent slave sub-problems on a lane level that can be solved in a decentralized manner. The control performance is verified via simulation at a four-arm signalized intersection. The simulation results show the joint control approach is flexible in incorporating multiple signal settings (such as cycle lengths and dual-ring design) and turning movements under different traffic demand levels and vehicle arrival rates. Furthermore, the benefits of the proposed control approach and computationally scalable algorithm in mean runtimes and performance metrics of travel delay, throughput, fuel consumption, and emission are revealed by comparison with three baselines.