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Recently the hybridization of microwave-frequency cavity modes with collective spin excitations has attracted large interest for the implementation of quantum computation protocols, which exploit the transfer of information among these two physical systems. Here we investigate the interaction among the magnetization precession modes of a small yttrium iron garnet sphere and the microwave electromagnetic modes, resonating in a tridimensional aluminum cavity. In the strong coupling regime, anticrossing features were observed corresponding to various magnetostatic modes, which were excited in a magnetically saturated sample. Time-resolved studies show evidence of Rabi oscillations, demonstrating coherent exchange of energy among photonic and magnon modes. To facilitate the analysis of the standing spin-wave patterns, we propose here a procedure, based on the introduction of a scaling variable. The resulting easier identification of magnetostatic modes can be exploited to investigate, control, and compare many-level hybrid systems in cavity- and opto-magnonics research.