= Solution
In four spacetime dimensions a <Weyl spinor> has <mass dimension> $3/2$, so each chiral current $J_L^\mu=\bar\psi_L\bar\sigma^\mu\psi_L$ or $J_R^\mu=\bar\psi_R\sigma^\mu\psi_R$ has dimension three. Since the interaction is $(gJ_L+g'J_R)^2$, dimensional homogeneity of the Lagrangian gives
$$
\boxed{[g]=[g']=-1}.
$$
Equivalently, the expanded <four-fermion interaction> has coefficients $g^2,gg',g'^2$ of mass dimension minus two.
Parity exchanges $J_L$ and $J_R$. The square is invariant precisely when the two same-chirality coefficients agree, $g^2=g'^2$, while the mixed coefficient is already symmetric. Therefore
$$
\boxed{g'=g\quad\text{or}\quad g'=-g}.
$$
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