We derive an efficient recursive procedure for the triangular factorization of strongly regular matrices with generalized displacement structure that includes, as special cases, a variety of previously studied classes such as Toeplitz-like and Hankel-like matrices. The derivation is based on combining a simple Gaussian elimination procedure with displacement structure, and leads to a transmission-like interpretation in terms of two cascades of first-order sections. We further derive state-space realizations for each section and for the entire cascades, and show that these realizations satisfy a generalized embedding result and a generalized notion of J-losslessness. The cascades turn out to have intrinsic blocking properties, which can be shown to be equivalent to interpolation constrains.