I'm not sure how we could find one in terms of the other. I know of the rule $$E[X] = E[X|A]P(A) + E[X|A^c]P(A^c)$$ but even if $E[X], P(A), P(A^c)$ are known, I don't see an immediate way to solve for a value of $E[X|A]$ that is not in terms of $E[X|A^c]$.
For example, take $E[X] = E[X|X < 1]P(X < 1) + E[X|X \geq 1]P(X \geq 1)$ for $X$ being a exponential random variable. I can reduce the equation to terms involving only $E[X|X < 1]$ and $E[X|X \geq 1]$, but am stuck here.