Is mean independence symmetric

Question

Consider two random variables X and Y

If X is mean independent of Y, is it necessarily true that Y is mean independent of X?

Answer 1

No: Consider the following four events with equal probabilities

 Y     X
+1    +2 
+1    -2
-1    +5
-1    -5

• $\mathbb E[X \mid Y=+1]= 0$ and $\mathbb E[X \mid Y=-1]= 0$
• $\mathbb E[Y \mid X=+2]= +1$ but $\mathbb E[Y \mid X=+5]= -1$