  In this paper, we study topological phases in a 2D photonic crystal with
broken time ($\mathcal{T}$) and parity ($\mathcal{P}$) symmetries by performing
calculations of band structures, Berry curvatures, Chern numbers, edge states
and also numerical simulations of light propagation in the edge modes.
Specifically, we consider a hexagonal lattice consisting of triangular
gyromagnetic rods. Here the gyromagnetic material breaks $\mathcal{T}$ symmetry
while the triangular rods breaks $\mathcal{P}$ symmetry. Interestingly, we find
that the crystal could host quantum anomalous Hall (QAH) phases with different
gap Chern numbers ($C_g$) including $|C_g| > 1$ as well as quantum valley Hall
(QVH) phases with contrasting valley Chern numbers ($C_v$), depending on the
orientation of the triangular rods. Furthermore, phase transitions among these
topological phases, such as from QAH to QVH and vice versa, can be engineered
by a simple rotation of the rods. Our band theoretical analyses reveal that the
Dirac nodes at the $K$ and $K'$ valleys in the momentum space are produced and
protected by the mirror symmetry ($m_y$) instead of the $\mathcal{P}$ symmetry,
and they become gapped when either $\mathcal{T}$ or $m_y$ symmetry is broken,
resulting in a QAH or QVH phase, respectively. Moreover, a high Chern number
($C_g = -2$) QAH phase is generated by gapping triply degenerate nodal points
rather than pairs of Dirac points by breaking $\mathcal{T}$ symmetry. Our
proposed photonic crystal thus provides a platform for investigating intriguing
topological phenomena which may be challenging to realize in electronic
systems, and also has promising potentials for device applications in photonics
such as reflection-free one-way waveguides and topological photonic circuits.
