I saw this post correlation between positive and negative part of a Random Variable and NCh's answer.
Here $X$ is a standard normal RV, $X^{+}, X^{-}$ are its positive and negative parts.
I find it interesting that $X^- = -\min(X, 0)$ is more commonly used, which would mean that $X^- \geq 0$ and $E[X^-] \geq 0$. What is the reasoning behind this instead of defining $X^ - = \min(X, 0) \leq 0$?
Depending on which definition you use your correlation/covariance will have opposite signs.
In addition, if we use the definition that $X^- = -\min(X, 0) \geq 0$. Then how can $E[X^+ X^-] = 0$? If we use the definition $X^- = \min(X, 0) \leq 0$, then we can make a symmetry argument that its expectation of the product is zero.
It does not matter which definition you use so long as the reader is clear. Similarly with the complex number $2+3i$, you could call the imaginary part $3i$ or $3$ but you need to be clear which you are using.
It may be more convenient in this case to have $X^-$ having the same distribution as $X^+$, which would justify using the non-negative version and $X=X^+ - X^-$
In either definition $X^+ \not=0 \implies X^- =0$ and $X^- \not=0 \implies X^+ =0$ and the product is always $0$
So $\mathbb P(X^+X^-=0)=1$ and $\mathbb E[X^+X^-]=0$