= Solution
The Weyl fields have odd <Grassmann parity>. With the given epsilon convention, write $\chi_\alpha=(a,b)$ and $\psi_\alpha=(c,d)$, where all four components anticommute. Raising an index gives $\chi^\alpha=(b,-a)$ and $\psi^\alpha=(d,-c)$. Hence
$$
(\chi\psi)=bc-ad,
\qquad(\psi\chi)=da-cb=-ad+bc.
$$
The minus sign from the antisymmetric epsilon tensor is canceled by exchanging the two Grassmann-odd components. Therefore the <Weyl spinor bilinear exchange identity> is
$$
\boxed{(\chi\psi)=(\psi\chi),\qquad C=1.}
$$
For commuting numerical spinors the exchange sign would instead be negative. The answer here uses fermionic Weyl fields, as appropriate to the supersymmetric model.
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