In this article, for multispacecraft systems (MSSs) with a directed complete communication topology and a time-varying virtual leader, an adaptive saturated attitude controller is proposed to achieve attitude consensus and attitude tracking under arbitrary initial attitude, mixed attitude constraints, input saturation, and external disturbances. First, considering the time-varying desired attitude provided by the virtual leader in a directed complete topology, an MSS attitude error function and an MSS attitude error dynamics based on SO(3) are developed. Next, an effective mixed potential function for the MSS on SO(3) is proposed for the static attitude-forbidden zones, the relative dynamic attitude-forbidden zones and the attitude-mandatory zones. Unlike the existing potential functions, the proposed mixed potential function is suitable for arbitrary initial attitude of the spacecraft in MSS, relaxing the restriction on the initial attitude associated with each static and dynamic attitude constraint zones. Then, an adaptive saturated attitude controller is designed to realize attitude consensus and tracking for the MSSs on SO(3) under arbitrary initial attitude, mixed attitude constraints, saturation constraints, and external disturbances. Finally, simulation results of an MSS with a time-varying virtual leader are demonstrated to illustrate the efficiency of the proposed attitude controller.