A method is introduced to perform simultaneous sparse dimension reduction on two blocks of variables. Beyond dimension reduction, it also yields an estimator for multivariate regression with the capability to intrinsically deselect uninformative variables in both independent and dependent blocks. An algorithm is provided that leads to a straightforward implementation of the method. The benefits of simultaneous sparse dimension reduction are shown to carry through to enhanced capability to predict a set of multivariate dependent variables jointly. Both in a simulation study and in two chemometric applications, the new method outperforms its dense counterpart, as well as multivariate partial least squares.