  Continuing our recent argument where we constructed a FNBWW-type spin-$1$
boson having opposite relative intrinsic parity to that of the associated
antiparticle, we now study eigenstates of the Charge Conjugation operator.
Based on the observation that if $\phi_{_{L}}(p^\mu)$ transforms as a $(0,\,j)$
spinor under Lorentz boosts, then $\Theta_{[j]}\,\phi_{_{L}}^\ast(p^\mu)$
transforms as a $(j,\,0)$ spinor (with a similar relationship existing between
$\phi_{_{R}}(p^\mu)$ and $\Theta_{[j]}\,\phi_{_{R}}^\ast(p^\mu)$; where $
\Theta_{[j]}\,{\bf J}\,\Theta_{[j]}^{-1}\,=\,-\,{\bf J}^\ast $ with
$\Theta_{[j]}$ the well known Wigner matrix involved in the operation of time
reversal) we introduce McLennan-Case type $(j,\,0)\oplus(0,\,j)$ spinors.
Relative phases between $\phi_{_{R}}(p^\mu)$ and
$\Theta_{[j]}\,\phi_{_{R}}^\ast(p^\mu)$, and
$\Theta_{[j]}\,\phi_{_{L}}^\ast(p^\mu)$ and $\phi_{_{L}}(p^\mu)$, turn out to
have physical significance and are fixed by appropriate requirements. Explicit
construction, and a series of physically relevant properties, for these spinors
are obtained for spin-$1/2$ and spin-$1$ culminating in the construction of a
new wave equation and introduction of Dirac-like and Majorana-like quantum
fields.
