Roberts orthogonality and Davis–Wielandt shell

Let A be a unital C*-algebra with the unit e. We consider the elements a in A which are Roberts orthogonal to the unit e. We obtain a characterization of this orthogonality in terms of the Davis–Wielandt shell of a and show that, for certain classes of elements of A, the Roberts orthogonality of a and e is equivalent to the symmetry of the numerical range of a with respect to the origin.