Time reversal of a real neutral chiral vector interaction ($B{\gamma }^{\mu \ast }{B}^{-1}=t_{\mu }{\gamma }^{\mu },B{\gamma }^{5\ast }{B}^{-1}={\gamma }^5$)

Let $B={\gamma }^{5}C$ with the unitary charge-conjugation matrix C, and $t_0=1,t_i=-1$. Hermitian-adjoint gamma identities give $C{\gamma }^{\mu \ast }{C}^{-1}=-{\gamma }^0{\gamma }^{\mu }{\gamma }^0$, and multiplication by ${\gamma }^5$ gives the displayed identities. An antiunitary operator conjugates the explicit i in ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$, which is essential to the second identity. Both vector and axial currents therefore have the same time-reversal component signs; their contraction with a similarly transforming neutral vector is invariant for real couplings.