Sorts Majorana spinor bilinears using the Majorana flip property, which for anti-commuting spinors takes the form $$\bar\psi_1 \Gamma_{r_1\cdots r_n}\psi_2 = \alpha \beta^n (-)^{\frac{1}{2}n(n-1)}\, \bar\psi_1 \Gamma_{r_1\cdots r_n}\psi_2\, .$$ Here $\alpha$ and $\beta$ determine the properties of the charge conjugation matrix, $${\cal C}^T = \alpha {\cal C}\,,\quad {\cal C}\Gamma_r {\cal C}^{-1} = \beta \Gamma_r^T\, .$$ Here is an example.
$$\displaystyle{}\bar{\chi} \Gamma_{m n} \psi$$
$$\displaystyle{}-\bar{\psi} \Gamma_{m n} \chi$$