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Apparently it is known that the signature of a product of manifolds is always 0. What is a canonical reference or a proof of that fact?

I am particularly interested in the signature of a product of 2 surfaces. Surfaces admit orientation-reversing diffeomorphisms. As is written in The signature of a product of surfaces by Qiaochu Yuan, this implies that the signature of the product of 2 surfaces is 0. Could you give the details of this implication?

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    What don't you understand about the proof given in the link? Also, a product of arbitrary surfaces can have nonzero signature; take $\mathbb{CP}^2\times \mathbb{CP}^2$. A product of oriented surfaces bounds and thus has signature $0$ (or for dimension reasons, or from explicitly computing the intersection form, or etc.). – 2017-02-12
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    I would like to understand the proof using the orientation-reversing diffeomorphisms of surfaces alluded to in the comment by Qiachu Yuan. – 2017-02-12
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    I'm not sure what you mean by that either. For a manifold $X$, the manifold $\bar{X}$ with opposite orientation has signature $\sigma(\bar{X}) = -\sigma(X)$, since the intersection form changes sign. So if $X$ is diffeomorphic to $\bar{X}$, then $\sigma(X) = 0$. – 2017-02-12
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    So you mean that signature is preserved by diffeomorphisms? I thought that it was preserved only by the orientation-preserving ones. – 2017-02-12
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    @Anastasia An orientation-reversing map $X \to X$ is the same as an orientation-preserving map $X \to \overline X$. – 2017-02-12
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    Thanks a lot! It was easier than I thought... – 2017-02-12

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