  In this study, we present the design of a photonic crystal (PC) structure
with a hexagonal lattice, where adjustments to the PC unit cell symmetry reveal
an all-angle self-collimation (SC) effect. By optimizing opto-geometric
parameters, such as the rotational angle of auxiliary rods and adjacent
distances, we analyze the SC property in detail, leveraging group velocity
dispersion (GVD) and third-order dispersion (TOD) characteristics. We also
investigate the relationship between symmetry properties and their influence on
dispersion characteristics. Through symmetry manipulation, we gain a
comprehensive understanding of the underlying mechanisms governing light
collimation and confinement in the proposed configurations. The PC structure
with a $C_1$ symmetry group exhibits all-angle SC effect within the range of
$a/\lambda=0.652$ and $a/\lambda=0.668$ normalized frequencies, with a
bandwidth of $\Delta\omega/\omega_c = 2.4\%$. Further breaking the symmetry,
transforming from $C_1$ to $C_2$ group symmetry, enhances the SC bandwidth to
$\Delta\omega/\omega_c = 6.5\%$ and reveals the perfect linear equi-frequency
contours (EFC) at two different frequency bands: all-angle SC between
$a/\lambda=0.616$ and $a/\lambda=0.656$ normalized frequencies in the \nth{4}
transverse magnetic (TM) band and between $a/\lambda=0.712$ and
$a/\lambda=0.760$ in the \nth{5} TM band. Additionally, we propose a
composite/hybrid PC structure resembling $C_2$ group symmetry, where two
auxiliary rods are replaced by rectangular photonic wires with the same
refractive index and width equal to the diameter of auxiliary rods. This hybrid
structure exhibits an all-angle SC effect with an operating bandwidth of
$\Delta\omega/\omega_c = 11.7\%$, displays near-zero GVD and TOD performance,
and offers enhanced robustness against potential fabrication precision issues.
