  The particular structure of Galileon interactions allows for
higher-derivative terms while retaining second order field equations for scalar
fields and Abelian $p$-forms. In this work we introduce an index-free
formulation of these interactions in terms of two sets of Grassmannian
variables. We employ this to construct Galileon interactions for mixed-symmetry
tensor fields and coupled systems thereof. We argue that these tensors are the
natural generalization of scalars with Galileon symmetry, similar to $p$-forms
and scalars with a shift-symmetry. The simplest case corresponds to linearised
gravity with Lovelock invariants, relating the Galileon symmetry to
diffeomorphisms. Finally, we examine the coupling of a mixed-symmetry tensor to
gravity, and demonstrate in an explicit example that the inclusion of
appropriate counterterms retains second order field equations.
