We study 4D N = 2 superconformal theories that arise from the compactification of 6D N = (2, 0) theories of type A(2N-1) on a Riemann surface C, in the presence of punctures twisted by a Z(2) outer automorphism. We describe how to do a complete classification of these SCFTs in terms of three-punctured spheres and cylinders, which we do explicitly for A(3), and provide tables of properties of twisted defects up through A(9). We find atypical degenerations of Riemann surfaces that do not lead to weakly-coupled gauge groups, but to a gauge coupling pinned at a point in the interior of moduli space.As applications, we study: i) 6D representations of 4D superconformal quivers in the shape of an affine/non-affine D-n Dynkin diagram, ii) S-duality of SU(4) and Sp(2) gauge theories with various combinations of fundamental and antisymmetric matter, and iii) realizations of all rank-one SCFTs predicted by Argyres and Wittig.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)