You misunderstood what I was saying. I was not suggesting that the KL divergence is symmetric. I was saying that it would be symmetric (and independent of the "

You misunderstood what I was saying. I was not suggesting that the KL divergence is symmetric. I was saying that it would be symmetric (and independent of the "truth" of a distribution) if it was interpreted as the quoted measure of "difference" between two distributions. So that proposed interpretation is wrong.