  Following Beurling's theorem the natural compressions of the multiplication
operator in the classical $L^2$ space are compressions to model spaces and to
their orthogonal complements. Two possibly different model spaces are
considered hence asymmetric truncated Toeplitz and asymmetric dual truncated
Toeplitz operators are investigated. The main purpose of the paper is to
characterize operators which intertwine compressions of the unilateral shift.
