This paper addresses the problem of the adaptation of a Gaussian Mixture Regression (GMR) to a new input distribution, using a limited amount of input-only examples. We propose a new model for GMR adaptation, called Joint GMR (J-GMR), that extends the previously published framework of Cascaded GMR (C-GMR). We provide an exact EM training algorithm for the J-GMR. We discuss the merits of the J-GMR with respect to the C-GMR and illustrate its performance with experiments on speech acoustic-to-articulatory inversion.