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^* ≡ρ_p/ρ_f and its relative moment-of-inertia I^* ≡I_p/I_f. This supersedes the current understanding that a critical mass density (m^*≈ 0.54) alone triggers the sudden onset of vigorous vibrations. Using over 144

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... nations 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 collapses into two scaling regimes demarcated by a transitional Strouhal number, St_t ≈ 0.17. St_t 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 2S (2-Single) vortex mode to a 2P (2-Pairs) vortex mode. Thus, auto-rotation can have a significant influence on the trajectories and wakes of freely rising isotropic bodies.