New high-frequency 5G and satellite communication systems require fully-metallic antennas and electromagnetic components. These components can be implemented with truncated versions of periodic structures. In order to achieve the desired performance of these future devices, it is of crucial importance to have a precise control of the propagation properties, i.e. the frequency dispersion behavior and stop-bands. Here, we demonstrate the potential use of higher symmetries to diminish the frequency dispersion of periodic structures and control the width of stop-bands with a new type of fully-metallic transmission line, which is loaded with holes on a twist-symmetric configuration. Simulated and experimental results confirm the intrinsic link between the propagation characteristics and the symmetries of a periodic structure. Additionally, we provide a definitive explanation of the recently discovered polar glide symmetry and its potential combination with twist symmetries to produce low-dispersive materials and reconfigurable stop-bands. The promising properties of these structures are demonstrated with a fully-metallic reconfigurable filter, which could be used for future high-frequency 5G and satellite communication systems.