An orthostochastic matrix is a bistochastic matrix $B$ such that $B_{ij}\equiv O_{ij}^2$ for some orthogonal matrix $O$.
Bistochastic matrices can be given a direct interpretation as describing stochastic processes: they describe how (specific types of) stochastic processes act on probability vectors.
Is there any intuition or more general understanding of the types of stochastic processes described by a bistochastic matrix that is also orthostochastic?