The unit quaternion is a pervasive representation of rigid-body attitude used for the design and analysis of feedback control laws. Because the space of unit quaternions constitutes a double cover of the rigid-body-attitude space, quaternion-based control laws are often-by design-inconsistent, i.e., they do not have a unique value for each rigid-body attitude. Inconsistent quaternion-based control laws require an additional mechanism that uniquely converts an attitude estimate into its quaternion representation; however, conversion mechanisms that are memoryless-e.g., selecting the quaternion having positive scalar component-have a limited domain where they remain injective and, when used globally, introduce discontinuities into the closed-loop system. We show-through an explicit construction and Lyapunov analysis-that such discontinuities can be hijacked by arbitrarily small measurement disturbances to stabilize attitudes far from the desired attitude. To remedy this limitation, we propose a hybrid-dynamic algorithm for smoothly lifting an attitude path to the unit-quaternion space. We show that this hybrid-dynamic mechanism allows us to directly translate quaternion-based controllers and their asymptotic stability properties (obtained in the unit-quaternion space) to the actual rigid-body-attitude space. We also show that when quaternion-based controllers are not designed to account for the double covering of the rigid-body-attitude space by a unit-quaternion parameterization, they can give rise to the unwinding phenomenon, which we characterize in terms of the projection of asymptotically stable sets. Finally, we employ the main results to show that certain hybrid feedbacks can globally asymptotically stabilize the attitude of a rigid body.