  We demonstrate the coexistence of pseudospin- and valley-Hall-like edge
states in a photonic crystal with $C_{3v}$ symmetry, which is composed of three
interlacing triangular sublattices with the same lattice constants. By tuning
the geometry of the sublattices, three complete photonic band gaps with
nontrivial topology can be created, one of which is due to the band inversion
associated with the pseudospin degree of freedom at the $\Gamma$ point and the
other two due to the gapping out of Dirac cones associated with the valley
degree of freedom at the $K, K'$ points. The system can support tri-band
pseudospin- and valley-momentum locking edge states at properly designed
domain-wall interfaces. Furthermore, to demonstrate the novel interplay of the
two kinds of edge states in a single configuration, we design a four-channel
system, where the unidirectional routing of electromagnetic waves against sharp
bends between two routes can be selectively controlled by the pseudospin and
valley degrees of freedom. Our work combines the pseudospin and valley degrees
of freedom in a single configuration and may provide more flexibility in
manipulating electromagnetic waves with promising potential for multiband and
multifunctional applications.
