We investigate a large class of infinitesimal, but fully nonlinear in the field, transformations of the Galileon and search for extended symmetries. The transformations involve powers of the coordinates x and the field π up to any finite order N. Up to quadratic order the structure of these symmetry transformations is the unique generalisation of both the infinitesimal version of the standard Galileon shift symmetry as well as a recently discovered infinitesimal extension of this symmetry. The

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... nly higher-order extensions of this symmetry we recover are ('Galileon dual' versions of) symmetries of the standard kinetic term.