This paper studies the fixed-time stability of coordinated attitude tracking control under consideration of an input amplitude constraint. The attitude dynamics is described by rotation matrix, which avoids the unwinding phenomenon. By designing and analyzing the proposed nonsingular terminal sliding mode surfaces (NTSMSs) step by step, a novel saturated NTSMS is finally established. This NTSMS has theoretical and practical advantages over other existing NTSMSs, such as its fixed-time convergence property and boundedness. Based on it, two fixed-time coordinated controllers are developed. The first one effectively decreases the amplitude of the control input, whereas the second prevents saturation. Both controllers keep the attitude synchronization task be finished in fixed-time and provide better performance than existing controllers. Specific expressions of the regions to which the system states to converge are obtained using Lyapunov stability theory. Simulation results testify the correctness of the proposed theorems.