Distributions where the KL-Divergence is symmetric I need to find two non-trivial examples for when the KL-Divergence happens to be symmetric for two distributions $P$ and $Q$, i.e.:
$$-\sum_{x\in\mathcal{X}} P(x) \log\left(\frac{Q(x)}{P(x)}\right)
=
-\sum_{x\in\mathcal{X}} Q(x) \log\left(\frac{P(x)}{Q(x)}\right). 
$$
I have already found the following example: Let $P$ and $Q$ be two Bernoulli distributed RVs. Then it must hold:
$$\sum_{x\in\mathcal{X}}-p_x \log\left(\frac{q_x}{p_x}\right)
-(1-p_x) \log\left(\frac{1-q_x}{1-p_x}\right)
=
\sum_{x\in\mathcal{X}}-q_x \log\left(\frac{p_x}{q_x}\right)
-(1-q_x) \log\left(\frac{1-p_x}{1-q_x}\right)
$$
This is true when $p_x = 1-q_x$ for all $x\in\mathcal{X}$. 
However, I am having a hard time to come up with a second example. Could you give me some hints?
 A: Consider any distribution (here, I am identifying a probability distribution with its probability mass function) $P$ supported on $[k]=\{1,2,3\dots,k\}$, and let $\pi\colon[k]\to[k]$ be an involution (i.e., a  permutation such that $\pi=\pi^{-1}$). Define $Q=P\circ\pi$; that is, for every $x\in [k]$, $Q(x) = P(\pi(x))$.
Then, since $\pi$ is a permutation, it is easy to check that
$$
\sum_{x\in[k]} P(x) \log P(x) = \sum_{x\in[k]} Q(x) \log Q(x) \tag{1}
$$
so that it suffices to show that 
$$
\sum_{x\in[k]} P(x) \log Q(x) = \sum_{x\in[k]} Q(x) \log P(x) \tag{2}
$$
But this is the case, as
$$\begin{align}
\sum_{x\in[k]} Q(x) \log P(x)
&= \sum_{x\in[k]} P(\pi(x)) \log P(x)
= \sum_{y\in[k]} P(y) \log P(\pi^{-1}(y))\\
&= \sum_{y\in[k]} P(y) \log P(\pi(y))
= \sum_{y\in[k]} P(y) \log Q(y)
\end{align}$$
where in the third equality we used $\pi^{-1}=\pi$.

Note that your example is a special case of the above, with $k=2$ and $\pi$ being the permutation which swaps the two elements of the domain.
