Note that the sign change depends on the Lorentzian signature of the metric because

but

so

i.e. the sign change is caused by the determinant of the metric.
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Note that the comment in the caption to figure 19.1 that
Quote:
"this sign change is absorbed by an effective index reversal."
is a potential source of confusion here. Index reversal has nothing to do with it, and in fact this is wrong, since the reversal of 4 indices is an even permutation that does not cause a sign change anyway.
See my post in the "Errors, typos, and corrections" forum
here.