  We predict that in a twisted homobilayer of transition-metal dichalcogenide
MoS$_2$, spin-orbit coupling in the conduction band states from $\pm K$ valleys
can give rise to moir\'{e} flat bands with nonzero Chern numbers in each
valley. The nontrivial band topology originates from a unique combination of
angular twist and local mirror symmetry breaking in each individual layer,
which results in unusual skyrmionic spin textures in momentum space with
skyrmion number $\mathcal{S} = \pm 2$. Our Hartree-Fock analysis further
suggests that density-density interactions generically drive the system at
$1/2$-filling into a valley-polarized state, which realizes a correlated
quantum anomalous Hall state with Chern number $\mathcal{C} = \pm 2$. Effects
of displacement fields are discussed with comparison to nontrivial topology
from layer-pseudospin magnetic fields.
