In this paper an approach to the design of robust global attitude tracking controllers for fully actuated rigid bodies is proposed. The challenge of simultaneously dealing with topological obstructions to global attitude tracking and with disturbances affecting the attitude dynamics is tackled by means of a hybrid hierarchical design that exploits the cascade structure of the underlying mathematical model. The proposed hierarchical strategy is based on an inner-outer loop paradigm comprising a dynamic control law for angular velocity tracking (inner loop) and a hybrid control law for attitude tracking (outer loop). By leveraging recent tools for the stability analysis of hybrid systems, we prove a robust global tracking property by assuming mild properties on the dynamics of the velocity feedback. We also discuss a few relevant examples satisfying these properties, encompassing harmonic disturbance compensators and conditional integrators, capable of rejecting unknown constant disturbances with an intrinsic anti-windup action.