We will describe several notions on the Galois side of the local Langlands correspondence that mirror well-known ones on the $p$-adic groups side. We will also explain how this can be used in the construction of an explicit arithmetical version of the correspondence. A typical example is provided by $G$-relevant cuspidal enhanced L-parameters, which are expected to correspond to the supercuspidal irreducible representations of the $p$-adic group $G$. Another examples are the Galois versions of the Bernstein blocks, their associated (possibly twisted) affine Hecke algebras, and certain elliptic pairings for representations of finite groups.