 Majorana Grassmann bilinear interchange

For Grassmann-odd spinors in two dimensions with $\bar{\psi }={\psi }^{T}C$, $C^T=-C$ and $({\gamma }^{\mu }{)}^{T}C=-C{\gamma }^{\mu }$, moving components past each other gives $\bar{\chi }ϵ=\overline{ϵ}\chi$ and $\bar{\chi }{\gamma }^{\mu }{\gamma }^{\nu }ϵ=\overline{ϵ}{\gamma }^{\nu }{\gamma }^{\mu }\chi$. The transpose of a two-gamma product reverses its order; the Grassmann sign and the antisymmetry of $C$ cancel. For $\delta \psi =B{\gamma }^{\mu }{\partial }_{\mu }Xϵ$, this also gives $\delta \bar{\psi }=-B\overline{ϵ}{\gamma }^{\mu }{\partial }_{\mu }X$. These identities fix the boundary variation in worldsheet supersymmetry; commuting spinors would not obey the same interchange rule.