  In this work, we build a covariant basis for operators acting on the
$(j,0)\oplus(0,j)$ Lorentz group representations. The construction is based on
an analysis of the covariant properties of the parity operator, which for these
representations transforms as the completely temporal component of a
symmetrical tensor of rank $2j$. The covariant properties of parity involve the
Jordan algebra of anti commutators of the Lorentz group generators which unlike
the Lie algebra is not universal. We make the construction explicit for
$j=1/2,1$ and $3/2$, reproducing well-known results for the $j=1/2$ case. We
provide an algorithm for the corresponding calculations for arbitrary $j$. This
covariant basis provides an inventory of all the possible interaction terms for
gauge and non-gauge theories of fields for these representations. In
particular, it supplies a single second rank antisymmetric structure, which in
the Poincar\'e projector formalism implies a single Pauli term arising from
gauge interactions and a single (free) parameter $g$, the gyromagnetic factor.
This simple structure predicts that for an elementary particle in this
formalism all multipole moments, $Q^{l}_{E}$ and $Q^{l}_{M}$, are dictated by
the complete algebraic structure of the Lorentz generators and the value of
$g$. We explicitly calculate the multipole moments, for arbitrary $j$ up to
$l=8$. Comparing with results in the literature we find that only the electric
charge and magnetic moment of a spin $j$ particle are independent of the
Lorentz representation under which it transforms, all higher multipoles being
representation dependent. Finally we show that the propagation of the
corresponding spin $j$ waves in a electromagnetic background is causal.
