Sub-Doppler laser spectroscopy of the A E-2(')<--X E-2(') band of Li-3 shows well-resolved and characteristic hyperfine structures of rovibronic transitions which can be explained in all detail by the interplay between the underlying Fermi-contact interaction, the symmetry species of the rovibronic states and the Coriolis coupling of rotational and pseudorotational motions. Ab initio calculated electron spin densities reproduce the observed line splittings to within 3%. In this Jahn-Teller distorted molecule magnetic equivalence of identical nuclei is strictly valid only for rovibronic states of A symmetry. However, it also holds for rovibronic states of E symmetry if they are of pure A vibronic origin, and it is enforced for many E states of pure E vibronic origin by Coriolis coupling. In contrast, those transitions which involve states with dominant K-c=0 rotation functions and/or with strong vibronic E/A mixing show significant changes in the hyperfine patterns due to magnetic inequivalence, sometimes called "hyperfine doubling."