Two signs of a Schwinger term in a commutator of fermionic currents

We consider central extensions of two dimensional abelian current algebra and Virasoro algebra and see that the sign of the Schwinger term changes if we arrange the Fourier modes of the fermion in the commutator in the order opposite to the normal. We argue that this is permissible in the case of fermions, but not in the case of bosons.