  The symmetry of the massive tower of fields in higher-dimensional Yang-Mills
theory compactified on a space-time of the form M_d x S^1 is clarified. The
transformations form a loop algebra, a class of Kac-Moody algebras. Since the
symmetry is spontaneously broken, vector fields "eat" Goldstone bosons and
acquire masses. The field of zero-mass mode can also become massive provided
that the field of the internal component develops a vacuum expectation value.
The relation between the "restoration" of the symmetry in massive modes and the
gauge transformation of the zero-mode vacuum field is discussed.
