Antiferromagnetic spin waves have been predicted to offer substantial functionalities for magnonic applications due to the existence of two distinct polarizations, the right-handed and left-handed modes, as well as their ultrafast dynamics. However, experimental investigations have been hampered by the field-immunity of antiferromagnets. Ferrimagnets have been shown to be an alternative platform to study antiferromagnetic spin dynamics. Here we investigate thermally excited spin waves in ferrimagnets across the magnetization compensation and angular momentum compensation temperatures using Brillouin light scattering. Our results show that right-handed and left-handed modes intersect at the angular momentum compensation temperature where pure antiferromagnetic spin waves are expected. A field-induced shift of the mode-crossing point from the angular momentum compensation temperature and the gyromagnetic reversal reveal hitherto unrecognized properties of ferrimagnetic dynamics. We also provide a theoretical understanding of our experimental results. Our work demonstrates important aspects of the physics of ferrimagnetic spin waves and opens up the attractive possibility of ferrimagnet-based magnonic devices. Right- and left-handed spin-wave modes are identified in ferrimagnets, and their dynamics are revealed.