I'm interested in the relationship between $$cov(X,Y)$$ and $$cov\left(E(X|Y),E(Y|X)\right).$$ In particular, can it occur that $cov(X,Y)>0$ but $cov(E(X|Y),E(Y|X))<0$?
In the case of $\begin{bmatrix}X_1\\X_2\end{bmatrix}\sim N\left(\begin{bmatrix}0\\0\end{bmatrix},\begin{bmatrix}1&\rho\\\rho&1\end{bmatrix}\right),$
$$cov(E(X_1|X_2),E(X_2|X_1))=E(\rho X_2\cdot\rho X_1)=\rho^2E(X_1X_2)=\rho^3.$$ So in this case the covariance of the conditional expectations always has the same sign as the original covariance, and smaller magnitude. Is this true in general?
One relation between the two can be arrived at using law of iterated expectation: \begin{align*} cov(E(X|Y), E(Y|X)) & = E[E(X|Y)\, E(Y|X)] - E[E(X|Y)]E[E(Y|X)]\\ & = E[E(X|Y)\, E(Y|X)] - E(X)E(Y)\\ & = E[E(X|Y)\, E(Y|X)] - (E(XY) - cov(X,Y) )\\ cov(E(X|Y), E(Y|X)) - cov(X,Y)& = E[E(X|Y)\, E(Y|X)] - E(XY)\,. \end{align*}
So $cov(E(X|Y), E(Y|X))$ can be larger than $cov(X,Y)$ if $E[E(X|Y)\, E(Y|X)]$ is larger than $E(XY)$.