  This paper analyzes thoroughly the dispersion and filtering features of
periodic holey waveguides in the millimeter-wave frequency range. Two
structures are mainly studied depending on the glide and mirror symmetries of
the holes. A parametric study of the dispersion characteristics of their unit
cells is carried out. Glide-symmetric holey waveguides provide a higher
propagation constant and a low dispersion over a wide frequency range regarding
hollow waveguides. This property is particularly useful for the design of
low-loss and low-dispersive phase shifters. We also demonstrate that
glide-symmetric holey waveguides are less dispersive than waveguides loaded
with glide-symmetric pins. Furthermore, we perform a Bloch analysis to compute
the attenuation constants in holey waveguides with mirror and broken glide
symmetries. Both configurations are demonstrated to be suitable for filter
design. Finally, the simulation results are validated with two prototypes in
gap-waveguide technology. The first one is a 180$^{o}$ phase shifter based on a
glide-symmetric holey configuration that achieves a flat phase shift response
over a wide frequency range (27.5\% frequency bandwidth). The second one is a
filter based on a mirror-symmetric holey structure with 20-dB rejection from 63
GHz to 75 GHz.
