  Under a rotation by an angle $\vartheta$, both the right- and left- handed
Weyl spinors pick up a phase factor ${\exp(\pm\, i \vartheta/2)}$. The upper
sign holds for the positive helicity spinors, while the lower sign for the
negative helicity spinors. For $\vartheta = 2\pi$ radians this produces the
famous minus sign. However, the four-component spinors are built from a direct
sum of the indicated two-component spinors. The effect of the rotation by
$2\pi$ radians on the eigenspinors of the parity - that is, the Dirac spinors
-- is the same as on Weyl spinors. It is because for these spinors the right-
and left- transforming components have the same helicity. And the rotation
induced phases, being same, factor out. But for the eigenspinors of the charge
conjugation operator, i.e. Elko, the left- and right- transforming components
have opposite helicities, and therefore they pick up opposite phases. As a
consequence the behaviour of the eigenspinors of the charge conjugation
operator (Elko) is more subtle: for $0<\vartheta<2\pi$ a self conjugate spinor
becomes a linear combination of the self and antiself conjugate spinors with
$\vartheta$ dependent superposition coefficients - and yet the rotation
preserves the self/antiself conjugacy of these spinors! This apparently
paradoxical situation is fully resolved. This new effect, to the best of our
knowledge, has never been reported before. The purpose of this communication is
to present this result and to correct an interpretational error of a previous
version.
