  We discuss a generalization of the Pauli-Gursey transformation, which is
motivated by the Autonne-Takagi factorization, to an arbitrary $n$ number of
generations of neutrinos using $U(2n)$ that defines general canonical
transformations and diagonalizes symmetric complex Majorana mass matrices in
special cases. The Pauli-Gursey transformation mixes particles and
antiparticles and thus changes the definition of the vacuum and C. We define C,
P and CP symmetries at each Pauli frame specified by a generalized Pauli-Gursey
transformation. The Majorana neutrinos in the C and P violating seesaw model
are then naturally defined by a suitable choice of the Pauli frame, where only
Dirac-type fermions appear with well-defined C, P and CP, and thus the C
symmetry for Majorana neutrinos agrees with the C symmetry for Dirac-type
fermions. This fully symmetric setting corresponds to the idea of Majorana
neutrinos as Bogoliubov quasi-particles. In contrast, the conventional direct
construction of Majorana neutrinos in the seesaw model, where CP is
well-defined but C and P are violated, encounters the mismatch of C symmetry
for Majorana neutrinos and C symmetry for chiral fermions; this mismatch is
recognized as the inevitable appearance of the singlet (trivial) representation
of C symmetry for chiral fermions.
