Charge conjugation and chirality

while I was reading about charge conjugation I found some (apparently) contradictory facts. For example Itzykson & Zuber says (page 153) "Up to a phase, $\cal C$ interchanges particles and antiparticlas with the same momentum, energy and helicity" while Zee (pag. 101) "You can easily convince yourself that the charged conjugated of a left handed field is a right handed field and viceversa" How can this be possible (in the case with $m=0$ when chirality coincides with helicity)?

In conclusion what is the handedness of the charge conjugated of a left handed field?

In order to convince myself I worked out two contradictory proofs: Let $U$ be the charge conjugation operator in Fock space, and $C$ the matrix that realizes charge conjugation on spinors: $\Psi^c = U^\dagger\Psi U = C\bar{\Psi}^t$, then:

1) $U^\dagger P_L\Psi U = P_L U^\dagger\Psi U = P_L C\bar{\Psi}^t$, (because $P_L$ is acting only upon creation and annihilation operators) and so here we proved that the charge conjugated of a left handed field is still left handed;

2) $U^\dagger P_L\Psi U = C\overline{(P_L\Psi)}^t = C\gamma^0 P_L\Psi^* = P_RC\bar{\Psi}^t$, (using Pauli-Dirac as well as Weyl representation of gamma matrices) and so here we proved that the charge conjugated of al left handed field is instead right handed.

Can you help me?

Note added: does a Majorana massless fermion exist?