Does spin direction of massless fermions require anti-particles?

If neutrinos were massless (which of course they are not), they would come in two types. The left-handed neutrino and the right-handed anti-neutrino.

Therefor to get an "up" state $|uparrowrangle$. Would this involve the superposition of a neutrino and an anti-neutrino $|leftarrowrangle + |rightarrowrangle$ ?

Therefor can you have a state point in all directions with superpositions of massless neutrinos and anti-neutrinos.

Is this different to the electron case?