= Time reversal of a real neutral chiral vector interaction
{title2=$B\gamma^{\mu*}B^{-1}=t_\mu\gamma^\mu,\quad B\gamma^{5*}B^{-1}=\gamma^5$}
Let $B=\gamma^5C$ with the unitary <charge-conjugation matrix> C, and $t_0=1,t_i=-1$. Hermitian-adjoint gamma identities give $C\gamma^{\mu*}C^{-1}=-\gamma^0\gamma^\mu\gamma^0$, 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.
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