We study attitude control of rigid bodies on quaternion coordinates under three mathematically different perspectives, depending on how the system dynamics are assumed to evolve. In the first case, we suppose that one equilibrium point is chosen a priori and a continuous controller is used under the assumption that the rigid body always spins in the same direction. In the second case, we relax the assumption that the sense of rotation is constant. Finally, a third scenario is considered in which hybrid (switching) control is used to choose the direction in which to spin, that is, both equilibria are continuously considered with regard to less energy consumption. It is showed that each of three scenarios must be treated in a different theoretical setting. A comparative study in simulations is also provided. (C) 2012 Elsevier Ltd. All rights reserved.