  We investigate a new class of scalar multi-galileon models, which is not
included in the commonly admitted general formulation of generalized
multi-galileons. The Lagrangians of this class of models, some of them having
already been introduced in previous works, are specific to multi-galileon
theories, and vanish in the single galileon case. We examine them in details,
discussing in particular some hidden symmetry properties which can be made
explicit by adding total derivatives to these Lagrangians. These properties
allow us to describe the possible dynamics for these new Lagrangians in the
case of multi-galileons in the fundamental representation of a SO(N) and SU(N)
global symmetry group, as well as in the adjoint representation of a SU(N)
global symmetry group. We perform in parallel an exhaustive examination of some
of these models, finding a complete agreement with the dynamics obtained using
the symmetry properties. Finally, we conclude by discussing what could be the
most general multi-galileon theory, as well as the link between scalar and
vector multi-galileon models.
