We present a new formulation of a staggered fermion on a D-dimensional lattice based on the SO(2D] Clifford algebra, which is naturally contained in the action. The action of a massless staggered fermion is invariant under discrete rotations and SO(2D) chiral and other discrete transformations. From the transformation properties of the fermion, we find two local meson operators (one scalar and one pseudoscalar), in addition to two standard meson operators.