  We present the rigorous derivation of covariant spin operators from a general
linear combination of the components of the Pauli-Lubanski vector. It is shown
that only two spin operators satisfy the spin algebra and transform properly
under the Lorentz transformation, which admit the two inequivalent
finite-dimensional representations for the Lorentz generators through the
complexification of the $SU(2)$ group. In case that the Poincar\'e group is
extended by parity operation, the spin operator in the direct sum
representation of the two inequivalent representations, called the new spin
distinguished from the Dirac spin, is shown to be equivalent to axial and
Hermitian spin operators for particle and antiparticle. We have shown that for
spin $1/2$, the Noether conserved current for a rotation can be divided into
separately conserved orbital and spin part for the new spin, unlike for the
Dirac spin. This implies that the new spin not the Dirac spin provides good
quantum observables.
