The dynamics of this third integrable case of classical rigid body theory is presented on several levels of abstraction, starting from real motion of a physical model and an analogous computer simulation of the Kovalevskaya equations of motion. The first abstraction involves the separation of a cyclic angular variable, i. e., the transition to a reduced description, and the introduction of a six-dimensional (gamma-iota)-phase with two Casimir constants. In the next step, relative equilibria are used to identify bifurcations between different topological types of three-dimensional energy surfaces. The third level is concerned with the foliation of these energy surfaces by invariant tori, and the identification of critical tori which mark bifurcations in the type of foliation. The tori are shown in various 3D projections and in homeomorphic deformations of the energy surfaces. The final step in the analysis uses the technique of Poincaré surfaces of section. A comprehensive survey on all possible motions is given in terms of animation series where all Poincaré sections for a given energy are shown in succession.