This paper develops a general framework for the description and analysis of Givens-weight algorithms applied to a rectangular matrix A with an irregular pattern of rank structured blocks. Givens-weight algorithms use transformations restricted to act on specific submatrices and have typically been described graphically. Here, we give an algebraic description of the fundamental operations used in such algorithms. We start with definitions of basic types of transformations and use those to build templates for the use of sequences of orthogonal transformations, such as in the computation of a rank structured QR decomposition. These templates have the advantage of making backward error analysis a simple matter, while also providing precise guidance on exactly when transformations should be applied in a practical implementation. We also describe how to use the rank structured QR decomposition to solve a linear system of equations by backward substitution, which differs from usual backward substitution by the use of transformations on the solution vector.