The optical properties of nanoparticles have attracted continuous attention owing to their high fundamental and applied importance across many disciplines. A recently emerged field of all-dielectric nanophotonics employs optically induced electric and magnetic Mie resonances in dielectric nanoparticles with a high refractive index. This property allows obtaining additional valuable degrees of freedom to control the optical responses of nanophotonic structures. Here we propose a conceptually distinct approach towards reaching optical resonances in dielectric nanoparticles. We show that, lacking conventional Mie resonances, low-index nanoparticles can exhibit a novel anisotropy-induced family of non-Mie eigenmodes. Specifically, we investigate light interactions with calcite and vaterite nanospheres and compare them with the Mie scattering by a fused silica sphere. Having close permittivities and the same dimensions, these particles exhibit significantly different scattering behavior owing to their internal structure. While a fused silica sphere does not demonstrate any spectral features, the uniaxial structure of the permittivity tensor for calcite and the non-diagonal permittivity tensor for vaterite result in a set of distinguishable peaks in scattering spectra. Multipole decomposition and eigenmode analysis reveal that these peaks are associated with a new family of electric and magnetic resonances. We identify magnetic dipole modes of ordinary, extraordinary and hybrid polarization as well as complex electric dipole resonances, featuring a significant toroidal electric dipole moment. As both vaterite and calcite are biominerals, naturally synthesized and exploited by a variety of living organisms, our results provide an indispensable toolbox for understanding and elucidating the mechanisms behind the optical functionalities of true biological systems.