  Using the recent parity-based construction of a covariant basis for operators
acting on the $(j,0)\oplus(0,j)$ representation of the HLG, we propose a
formalism for the description of high spin matter fields, based on the
projection over subspaces of well-defined parity. We identify two possibilities
for the projection, on-shell and off-shell projection. For all $j$ except for
$j=1/2$, we find that the projection does not completely fix the properties of
the interacting theory. This freedom is related to the fact that the covariant
form of parity can be written in terms of one of the symmetric traceless
tensors in the covariant basis and in general allows for a free magnetic dipole
term in the lagrangian. We gauge the theory and construct the charge
conjugation operator. In the case of bosons, the parity invariant subspaces are
also invariant under charge conjugation and time reversal and the formulation
of a quantum field theory can be done using only these subspaces. As a first
exhaustive example we work out the electrodynamics for $j=1$ matter bosons,
rewrite the theory in terms of an antisymmetric tensor field and compare our
results with existing formalisms in the literature. We find that there are
three essentially different formalisms: i) formalisms equivalent to the
on-shell parity projection, ii) formalisms equivalent to the off-shell parity
projection and iii) the Poincar\'e projector formalism which describes a
degenerate parity doublet. We perform a chiral decomposition of these theories
and show that chiral symmetry can be realized linearly only for the theory
based on the on-shell projection. Chiral symmetry forbids mass and anomalous
magnetic dipole terms and in general admits six self-interaction terms. We
conclude that this is the appropriate framework to attempt the incorporation of
spin 1 matter bosons in chiral theories like the standard model.
