Variance of sum of normal and half-normal

Question

Let $X\sim \mathcal{N}(0, \sigma)$. Define $Z = \max(X, 0) = (X + \vert X \vert)$/2. Given that $\vert X \vert$ follows a half-normal distribution, we have: $$\mathbb{E}[Z] = \frac{0 + \sigma\sqrt{2/\pi}}{2} = \sigma / \sqrt{2\pi}.$$ We further have: $$\mathrm{Var}(Z) = \frac{\mathrm{Var}(X) + \mathrm{Var}(\vert X \vert) + 2\mathrm{Cov}(X, \vert X \vert)}{4}=\frac{\sigma^2 + \sigma^2(1-2/\pi^2) + 2\mathrm{Cov}(X, \vert X \vert)}{4}.$$ My question is: what is the covariance of $X$ and $\vert X \vert$?

Answer 1

$E[X|X|]=0$ by symmetry. So the covariance of $X$ and $|X|$ is $0-(EX)(E|X|)=0$.