= Solution
A <Grassmann variable> is an odd generator of a <Grassmann algebra>: $\eta_i\eta_j=-\eta_j\eta_i$, so $\eta_i^2=0$. For one generator, any function is $f(\eta)=a+b\eta$. The <Berezin integral> is the linear operation
$$
\boxed{\int d\eta\,1=0,\qquad\int d\eta\,\eta=1},
\qquad \int d\eta\,f(\eta)=b.
$$
Thus integration extracts a coefficient, rather than assigning a length or volume. For many generators it extracts the coefficient of the highest-degree monomial with the sign fixed by the order of the measure. Odd coefficients and <Grassmann derivatives> must retain their order; exchanging two odd objects changes the sign.
This operation is invariant under odd translations, because a translation only changes terms of lower degree. For an invertible ordinary matrix $S$ and $\eta=S\xi$, the <Grassmann change-of-variables formula> is
$$
D\eta=(\det S)^{-1}D\xi.
$$
The inverse <Jacobian determinant>, rather than the ordinary commuting-variable Jacobian, compensates for the factor $\det S$ multiplying the top monomial. Integration agrees with the appropriate ordered <Grassmann derivatives>, but the orientation must be specified when combining barred and unbarred variables.
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