  A well-established organisational principle for Argyres--Douglas-type
$\mathcal{N}=2$ superconformal field theories in four dimensions is to
characterise such theories by the data defining a(n irregular) Hitchin system
on $\mathbb{CP}^1$. The dictionary between Hitchin system data and various
features of the corresponding SCFT has been studied extensively, but the
overall structure of the resulting space of SCFTs still appears quite
complicated. In this work, we systematically delineate a variety of
simplifications that arise within this class of constructions due to several
large classes of isomorphisms between SCFTs associated with inequivalent
Hitchin system data (and their exactly marginal gaugings). We restrict to the
most studied class of theories, namely the type $A$ theories without outer
automorphism twists.
