The KL divergence between two distribution $p$ and $q$ is defined as $$ D( q \| p)\int q(x)\log \frac{q(x)}{p(x)} dx $$ and is known to be asymmetry: $D(q\|p)\neq D(p\|q)$.

If we fix $p$ and try to find a distribution $q$ among a class $E$ that minimize the KL distance, it is also known that minimizing $D( q \| p)$ will be different from minimizing $D( p \| q)$, e.g., https://benmoran.wordpress.com/2012/07/14/kullback-leibler-divergence-asymmetry/.

It is not clear which one to be optimized for a better approximation, although in many application we minimize $D(q\|p)$.

My question is that, when will the solution be the same

$$ \underset{q\in E}{\operatorname{argmin}} D(q\|p) = \underset{q\in E}{\operatorname{argmin}} D(p\|q)? $$

For instance, if we take $E$ as the class of all Gaussian distribution, is there a condition on $p$ so that minimizing these two will lead to the same minimizer?