In the regression model $$y= x'\beta + u, \quad x = (1, x_2,...,x_K)$$ with $$E[u |x]=0,$$
we know that it implies: $E [u] = 0$ and $cov(u,x_j)=0$, for $j=1,...,K$.
I think that the reciprocal is not true. Thinking geometrically, they seem equivalent, but I can't come up with a counterexample!
Do you have a counterexample?
Defining $\langle X,Y\rangle= E[XY]$ we have that if $X$ or $Y$ is such that $E[X]=0$ or $E[Y]=0$, we have: $$\langle X,Y\rangle=cov(X,Y)$$ So, if $cov(u,x_j)=0$, I have orthoganility: $\langle u,x_j\rangle=0$ for all $j=1,...,k$. This seems to imply that the projection of $u$ onto $1,x_2,...,x_k$ is $0$, i.e. $E[u|x]=0$.