We provide a novel interpretation of the dual of support vector machines (SVMs) in terms of scatter with respect to class prototypes and their mean. As a key contribution, we extend this framework to multiple classes, providing a new joint Scatter SVM algorithm, at the level of its binary counterpart in the number of optimization variables. This enables us to implement computationally efficient solvers based on sequential minimal and chunking optimization. As a further contribution, the primal problem formulation is developed in terms of regularized risk minimization and the hinge loss, revealing the score function to be used in the actual classification of test patterns. We investigate Scatter SVM properties related to generalization ability, computational efficiency, sparsity and sensitivity maps, and report promising results.