  We study $4D$ $\mathcal{N}=2$ superconformal field theories that arise as the
compactification of the six-dimensional $(2,0)$ theory of type $E_6$ on a
punctured Riemann surface in the presence of $\mathbb{Z}_2$ outer-automorphism
twists. We explicitly carry out the classification of these theories in terms
of three-punctured spheres and cylinders, and provide tables of properties of
the $\mathbb{Z}_2$-twisted punctures. An expression is given for the
superconformal index of a fixture with twisted punctures of type $E_6$, which
we use to check our identifications. Several of our fixtures have Higgs
branches which are isomorphic to instanton moduli spaces, and we find that
S-dualities involving these fixtures imply interesting isomorphisms between
hyperK\"ahler quotients of these spaces. Additionally, we find families of
fixtures for which the Sommers-Achar group, which was previously a Coulomb
branch concept, acts non-trivially on the Higgs branch operators.
