In the description of the Langlands correspondence for $\mathbb{Q}_p$, we consider admissible representations of $G(\mathbb{Q}_p)$ for $G$ a reductive group defined over $\mathbb{Q}_p$, and admissible Langlands parameters $\phi: W \to ^LG$, where $W$ denotes the absolute Weil group of $\mathbb{Q}_p$. In particular, to each irreducible admissible representation $\pi$ of $G(\mathbb{Q}_p)$, we (to some extent conjecturally) assign an $L$-parameter $\pi_{\phi}$. The map $\pi \mapsto \pi_{\phi}$ satisfies a number of compatibility conditions and is finite to one. The set theoretic fiber of an $L$-parameter is called an $L$-packet and denoted $\Pi_{\phi}$. Various descriptions of the contents of $\Pi_{\phi}$ have been suggested (in particular in the tempered case), typically involving some variant of the set of irreducible representations of the centralizer of the image of $\phi$, denoted $S_{\phi}$.

My question asks for an intuitive (or otherwise) explanation of why representations of this centralizer group should be involved in parametrising the contents of the corresponding $L$-packet.