Difference Sturm–Liouville operators in imaginary direction and bispectrality

We consider self-adjoint difference operators in $L^2$ on real line with shifts by the imaginary unit. A domain of defineteness of such operators consists of functions holomorphic in a strip. Now we know a zoo of explicit spectral decompositions of such operators. It appears that in all known cases inverse unitary operators provide us explicit spectral decompositions of another Sturm–Liouville operators (differential, difference, or difference in imaginary direction). There are known also some explicitly solvable problems for commuting families of such operators.