Curious patterns of IR symmetry enhancement

We study several cases of IR enhancements of global symmetry in four dimensions. In particular, we consider a sequence of $Spin(n+4)$ supersymmetric gauge theories ($8\geq n\geq 1$) with $n$ vectors and spinor matter with $32$ components. We show that the subgroup of the flavor symmetry of these theories rotating the matter in the spinor representations in the UV, when proper gauge singlet fields are added, enhances to the commutant of $SU(2)$ in $E_{9-n}$. We discuss several other interesting cases of enhanced symmetries and the interplay between symmetry enhancement and self-duality. We also make some observations about possible interconnections between chiral ring relations and symmetry enhancement. Finally, we conjecture relations of the discussed models to compactifications of certain conformal matter models in six dimensions on tori. The conjecture is based on deriving a relation between five dimensional models with $Spin$ gauge groups and conformal theories in six dimensions. As a by product of our considerations we discover a new instance of a simple self-duality of a theory with an $SU(6)$ gauge group.