The presence of clusters in a data set is sometimes due to the existence of certain relations among the measured variables which vary depending on some hidden factors. In these cases, observations could be grouped in a natural way around linear and nonlinear structures and, thus, the problem of doing robust clustering around linear affine subspaces has recently been tackled through the minimization of a trimmed sum of orthogonal residuals. This “orthogonal approach” implies that there is no privileged variable playing the role of response variable or output. However, there are problems where clearly one variable is wanted to be explained in terms of the other ones and the use of vertical residuals from classical linear regression seems to be more advisable. The so-called TCLUST methodology is extended to perform robust clusterwise linear regression and a feasible algorithm for the practical implementation is proposed. The algorithm includes a “second trimming” step aimed to diminishing the effect of leverage points.