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Polaritons are a hybrid class of quasiparticles originating from the strong and resonant coupling between light and matter excitations. Recent years have witnessed a surge of interest in new polariton types, arising from directional, long-lived material resonances, and leading to extreme optical anisotropy that enables new regimes of nanoscale, highly confined light propagation. Although such exotic propagation features may also in principle be achieved by using carefully designed metamaterials, it has recently been realized that they can naturally emerge when coupling infrared light to directional lattice vibrations — phonons — in polar crystals. Interestingly, a reduction in crystal symmetry increases the directionality of optical phonons and the resulting anisotropy of the response, which in turn enables new polaritonic phenomena, such as hyperbolic polaritons with highly directional propagation, ghost polaritons with complex-valued wavevectors, and shear polaritons with strongly asymmetric propagation features. In this Review, we develop a critical overview of recent advances in the discovery of phonon polaritons in low-symmetry crystals, highlighting the role of broken symmetries in dictating the polariton response and associated nanoscale light propagation features. We also discuss emerging opportunities for polaritons in lower-symmetry materials and metamaterials, with connections to topological physics and the possibility of using anisotropic nonlinearities and optical pumping to further control their nanoscale response.