  A Poincar\`{e} invariant, local scalar field theory in which the Lagrangian
and the equation of motion contain only up to second-order derivatives of the
fields is called generalized Galileon. The covariant version of it in four
dimensions is called Horndeski theory, and has been vigorously studied in
applications to inflation and dark energy. In this paper, we study a class of
multi-field extensions of the generalized Galileon theory. By imposing shift
and $SO(N)$ symmetries on all the currently known multi-Galileon terms in
general dimensions, we find that the structure of the Lagrangian is uniquely
determined and parameterized by a series of coupling constants. We also study
tensor perturbation in the shift-symmetric $SO(3)$ multi-Galileon theory in
four dimensions. The tensor perturbations can obtain a mass term stemming from
the same symmetry breaking pattern as the solid inflation. We also find that
the shift-symmetric $SO(3)$ multi-Galileon theory gives rise to new cubic
interactions of the tensor modes, suggesting the existence of a new type of
tensor primordial non-Gaussianity.
