  We prove that the $SU(3)_C\otimes SU(2)_L \otimes SU(3)_R\otimes U(1)_X$
(3-2-3-1) gauge model always contains a matter parity $W_P=(-1)^{3(B-L)+2s}$ as
conserved residual gauge symmetry, where $B-L=2(\beta T_{8R}+X)$ is a
$SU(3)_R\otimes U(1)_X$ charge. Due to the non-Abelian nature of $B-L$, the
$W$-odd and $W$-even fields are actually unified in gauge multiplets. We
investigate two viable versions for dark matter according to
$\beta=\pm1/\sqrt{3}$, where the dark matter candidates can be fermion, scalar,
or vector fields. We figure out the parameter spaces in the allowed regions of
the relic density and direct detection cross-sections. Additionally, we examine
the neutrino masses induced by the seesaw mechanism along with associated
lepton flavor violation processes. The new gauge boson searches at the LEPII
and LHC are discussed.
