Some equation involving $cov(X,Y)$ and $E(X|Y)$

I am trying to find some math equation envolving $cov(X,Y)$ and the conditional expectation $E(X|Y)$.

I'm willing to put some hypotheses. For example, if $E(X)=0$ or $E(Y)=0$, then $cov(X,Y) = E(XY)$. Thus, I am looking for some relation between $E(XY)$ and $E(X|Y)$. The closest I found was the Law of total covariance:

$$\operatorname{cov}(X,Y)=\operatorname{E}(\operatorname{cov}(X,Y \mid Z))+\operatorname{cov}(\operatorname{E}(X\mid Z),\operatorname{E}(Y\mid Z))$$

But this isn't exactly what I'm looking for.

Does anyone know any classic formula?

Using conditioning on $Y$, we can get: $$ E[XY] = E[E[XY|Y]] = E[YE[X|Y]]$$

This also leads to:

$${\rm cov}(E[X\mid Y],Y) = {\rm cov}(X,Y) $$