In this paper, 1-D periodic structures possessing higher symmetries are proposed and investigated in terms of their dispersion properties. The proposed structures are coaxial lines with coaxial rings periodically loaded on their inner or outer conductors. The higher symmetries, namely, glide and twist symmetries, are obtained by performing an additional geometrical operation within the unit cell of the periodic structure. We demonstrate that the propagating modes exhibit a lower frequency dispersion in higher symmetric coaxial lines. Moreover, the conventional stopbands of periodic structures at their Brillouin zone boundaries can be controlled by breaking the higher symmetry or changing the order of the twist symmetry. A circuit-based analytical method is proposed to calculate the dispersion diagram of the glide-symmetric coaxial lines. The results are validated with a full-wave simulation. Moreover, several prototypes of the twist-symmetric coaxial lines are manufactured and measured. A remarkable agreement is achieved between the measurements and simulations, validating the theoretical results. The proposed structures find potential applications in leaky-wave antennas and fully metallic reconfigurable filters and phase shifters.