I know that mutual information is the Kullback-Leibler divergence between $p(x,y)$ and $p(x)p(y)$. But mutual information is also described as the amount of entropy lost (or, in another sense, the information gained) about $X$ by virtue of knowing about $Y$.

Without just looking at the formula, it's not clear why the additional info about $X$ due to having $Y$ would be the same as the additional information about $Y$ due to having $X$. It's not immediately obvious that predicting $Y$ from the behavior of $X$ is just as easy as predicting $X$ from the behavior of $Y$. While "mutual" is in the name, mutual information is described in terms of learning about $X$ using Y, and so in the same way that e.g. KL divergence (which is described in terms of describing $X$ using $Y$) is asymmetric, the intuition for me would have been that "mutual" information is asymmetric too.

Is there any intuition here without just looking at the formula for $I(X;Y)$?