  In non-centrosymmetric metals, spin-orbit coupling (SOC) induces
momentum-dependent spin polarization at the Fermi surfaces. This is exemplified
by the valley-contrasting spin polarization in monolayer transition metal
dichalcogenides (TMDCs) with in-plane inversion asymmetry. However, the valley
configuration of massive Dirac fermions in TMDCs is fixed by the graphene-like
structure, which limits the variety of spin-valley coupling. Here, we show that
the layered polar metal BaMn$X_2$ ($X =$Bi, Sb) hosts tunable
spin-valley-coupled Dirac fermions, which originate from the distorted $X$
square net with in-plane lattice polarization. We found that in spite of the
larger SOC, BaMnBi$_2$ has approximately one-tenth the lattice distortion of
BaMnSb$_2$, from which a different configuration of spin-polarized Dirac
valleys is theoretically predicted. This was experimentally observed as a clear
difference in the Shubnikov-de Haas oscillation at high fields between the two
materials. The chemically tunable spin-valley coupling in BaMn$X_2$ makes it a
promising material for various spin-valleytronic devices.
