We simulate two dimensional QED with two degenerate Wilson fermions and plaquette gauge action. As a consequence of the Mermin-Wagner theorem, in the continuum limit chiral symmetry is realized a la Wigner. This property affects also the size of the cutoff effects. That can be understood in view of the fact that the leading lattice artifacts are described, in the continuum Symanzik effective theory, by chirality breaking terms. In particular, vacuum expectation values of non-chirality-breaking operators are expected to be O(a) improved in the chiral limit. We provide a numerical confirmation of this expectation by performing a scaling test.