Does zero correlation imply mean independence in at least one direction?

Question

Any example of two random variables $X$ and $Y$ with correlation $\rho(X,Y)=0$ seems to satisfy that either $X$ is mean independent of $Y$, $\mathbb{E}[Y\mid X]=E[Y]$, or vice versa, $\mathbb{E}[X\mid Y] = E[X]$. Does anyone know of a joint distribution in which both $X$ and $Y$ are mean dependent of each other, but their respective correlation is also zero?

Answer 1

I will lazily and shamelessly take this from Wikipedia, which does a fine job summing this up.

Answer 2

If $X$ has standard normal distribution then the covariance of $X$ and $X^2$ is $0$. But they are not mean independent because $E(X^2\mid X) = X^2 \neq EX^2$.