At the cellular scale, blood fluidity and mass transport depend on the dynamics of red blood cells in blood flow, specifically on their deformation and orientation. These dynamics are governed by cellular rheological properties, such as internal viscosity and cytoskeleton elasticity. In diseases in which cell rheology is altered genetically or by parasitic invasion or by changes in the microenvironment, blood flow may be severely impaired. The nonlinear interplay between cell rheology and flow may generate complex dynamics, which remain largely unexplored experimentally. Under simple shear flow, only two motions, "tumbling" and "tank-treading," have been described experimentally and relate to cell mechanics. Here, we elucidate the full dynamics of red blood cells in shear flow by coupling two videomicroscopy approaches providing multidirectional pictures of cells, and we analyze the mechanical origin of the observed dynamics. We show that contrary to common belief, when red blood cells flip into the flow, their orientation is determined by the shear rate. We discuss the "rolling" motion, similar to a rolling wheel. This motion, which permits the cells to avoid energetically costly deformations, is a true signature of the cytoskeleton elasticity. We highlight a hysteresis cycle and two transient dynamics driven by the shear rate: an intermittent regime during the "tank-treading-to-flipping" transition and a Frisbee-like "spinning" regime during the "rolling-to-tank-treading" transition. Finally, we reveal that the biconcave red cell shape is highly stable under moderate shear stresses, and we interpret this result in terms of stress-free shape and elastic buckling.