Consider a (sub)-stochastic matrix $P$, and the associated Markov chain $X$ with \begin{align*} \mathbf P [X_n =y | X_0 = x] = P_{xy}^n. \end{align*}
Suppose we have the condition $P^T P = P P^T$, i.e. the transition matrix is normal. Is there a probabilistic interpretation of this?