quantum field theory - Is $CP$ instead of $C$ responsible for changing a particle to its antiparticle?

The charge conjugation operator $C$ reverses the charge of a state. But it may or may not convert a particle to its antiparticle. For example, consider a neutrino which is charge-neutral and left-handed while its antiparticle is also charge-neutral but right-handed. Therefore, charge conjugation is not sufficient to produce the antineutrino from a neutrino but CP is. $P$ is responsible for changing the chirality.

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   So is it not $CP$ instead of $C$ that is responsible for changing a particle to its antiparticle?

   
   


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   Is this fact related to the questions here and here?