On the physical parity transformation and antiparticles

It was argued a long time ago by T.D. Lee and G.C. Wick that there is a basis where
the usual CP (charge-parity) transformation becomes a parity transformation. Indeed, at
the quantum level all fields are real representations of the group of symmetries, because
CP (charge-parity) is a linear transformation. From a canonical quantization point of view,
there are no complex-conjugate representations, and all particles are their own antiparticles.
The Poincare group is the quotient group of the group of symmetries by the normal subgroup
group of internal symmetries (which may include the charge conjugation). Thus, CP is a
candidate for a physical parity transformation (included in the Poincare group). We check
explicitly what this implies for the Standard Model and discuss the implications for Left-
Right symmetric models and models featuring a CP order-4 symmetry.