  We study certain linear and antilinear symmetry generators and involution
operators associated with pseudo-Hermitian Hamiltonians and show that the
theory of pseudo-Hermitian operators provides a simple explanation for the
recent results of Bender, Brody and Jones (quant-ph/0208076) on the
CPT-symmetry of a class of PT-symmetric non-Hermitian Hamiltonians. We present
a natural extension of these results to the class of diagonalizable
pseudo-Hermitian Hamiltonians H with a discrete spectrum. In particular, we
introduce generalized parity (${\cal P}$), time-reversal (${\cal T}$), and
charge-conjugation (${\cal C}$) operators and establish the ${\cal PT}$- and
${\cal CPT}$-invariance of H.
