In this Letter, we study the motion and wake patterns of freely rising and falling cylinders in quiescent fluid. We show that the amplitude of oscillation and the overall system dynamics are intricately linked to two parameters: the particle's mass density relative to the fluid m∗≡ρp/ρf and its relative moment of inertia I∗≡Ip/If. This supersedes the current understanding that a critical mass density (m∗≈0.54) alone triggers the sudden onset of vigorous vibrations. Using over 144 combinations of m∗ and I∗, we comprehensively map out the parameter space covering very heavy (m∗>10) to very buoyant (m∗<0.1) particles. The entire data collapse into two scaling regimes demarcated by a transitional Strouhal number Stt≈0.17. Stt separates a mass-dominated regime from a regime dominated by the particle's moment of inertia. A shift from one regime to the other also marks a gradual transition in the wake-shedding pattern: from the classical two-single (2S) vortex mode to a two-pair (2P) vortex mode. Thus, autorotation can have a significant influence on the trajectories and wakes of freely rising isotropic bodies.