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A<sc>bstract</sc> This paper discusses a framework to parametrize and decompose operator matrix elements for particles with higher spin (j> 1/2) using chiral representations of the Lorentz group, i.e. the (j, 0) and (0,j) representations and their parity-invariant direct sum. Unlike traditional approaches that require imposing constraints to eliminate spurious degrees of freedom, these chiral representations contain exactly the 2j+ 1 components needed to describe a spin-jparticle. The central objects in the construction are thet-tensors, which are generalizations of the Pauli four-vectorσ^μfor higher spin. For the generalized spinors of these representations, we demonstrate how the algebra of thet-tensors allows to formulate a generalization of the Dirac matrix basis for any spin. For on-shell bilinears, we show that a set consisting exclusively of covariant multipoles of order 0 ≤m≤ 2jforms a complete basis. We provide explicit expressions for all bilinears of the generalized Dirac matrix basis, which are valid for any spin value. As a byproduct of our derivations we present an efficient algorithm to compute thet-tensor matrix elements. The formalism presented here paves the way to use a more unified approach to analyze the non-perturbative QCD structure of hadrons and nuclei across different spin values, with clear physical interpretation of the resulting distributions as covariant multipoles.