We continue research by A. Cellina who formulated the variational version of the Strong Maximum Principle (SMP) for a functional rotationally symmetric and depending on the gradient. He proved that for such type of functionals SMP is valid if and only if the real function f(·) is strict convex and smooth at the origin. Generalizing the symmetry assumption we prove, in particular, that the same conditions are necessary and sufficient for validity of SMP for a more general functional, symmetric with respect to a gauge function associated to a closed convex bounded set F with zero in its interior. On the other hand, by using some a priori local estimates we establish a generalized version of SMP in the case when f (·) is no longer supposed to be strictly convex at the origin.