 Worldsheet chirality and propagation direction

In signature $(-,+)$, choose real gamma matrices ${\gamma }^0=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$, ${\gamma }^1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and ${\gamma }_{\ast }=-{\gamma }^0{\gamma }^1$. The Dirac equation then gives $({\partial }_{\tau }+{\partial }_{\sigma }){\psi }_{+}=0$ and $({\partial }_{\tau }-{\partial }_{\sigma }){\psi }_{-}=0$. Positive chirality is right-moving and negative chirality is left-moving in this orientation. Reversing the definition of ${\gamma }_{\ast }$ exchanges the labels without changing the physics.