Does anyone know about a statistical divergence of this type? \begin{equation} \text{D}(P||Q) = \frac{1}{2} \left[\text{KL}(M||P) + \text{KL}(M||Q)\right] \end{equation} where $M = \frac{1}{2} [P+Q]$.

This divergence is very similar to the Jensen-Shannon Divergence $\text{D}(P||Q) = \text{KL}(P||M) + \text{KL}(Q||M)$ but where the distributions in the argument of the statistical divergence appear in the second argument of the KL divergences.

I am interested in knowing if such divergence exists in the literature and to properties of such divergence. Thanks!