Independence between two events, $A$ and $B$, is a symmetric relation, that is, if $P(A \mid B) = P(A)$, then $P(B \mid A) = P(B)$. The proof is very simple and can be found at the ProofWiki.

Intuitively, when I think about dependence, I think about relations like $A = 2*B$. In this case, $A$ clearly depends on $B$ (it is always the double of $B$). However, in this case, the equation doesn't say that the value of $B$ depends on the value of $A$. Indeed, if that was the case, this would be a circular definition. So, in general, when I think about $A$ being dependent on $B$, I don't really think about $B$ being also dependent on $A$ (even in our every day life, this is not usually the case). However, in this case, we are talking about dependence and not independence. Nonetheless, if event $A$ is not dependent on $B$, then event $A$ is independent of $B$ (trivial). But does this imply that the dependence relation is also symmetric, that is, if $P(A \mid B) \neq P(A)$, then $P(B \mid A) \neq P(B)$?

In general, if dependence is also symmetric, how do we intuitively understand that this is the case? Also, in general, how do we intuitively understand that independence is symmetric?