Covariance of X with the square of a correlated normal

Suppose $(X,Y)$ is bivariate normal, each coordinate standard normal, with correlation $\rho$. What is $\operatorname{Cov}(X,Y^2)$?

$0$. Since $\mathbb{E}[X]=0$, the covariance equals $\mathbb{E}[XY^2]$. The pair $(-X,-Y)$ has the same joint law as $(X,Y)$, and $XY^2$ changes sign under this flip, so its expectation is zero for every $\rho$.