We study 4D N = 2 superconformal field theories that arise from the compactification of 6D N = (2, 0) theories of type D-N on a Riemann surface, in the presence of punctures twisted by a Z(2) outer automorphism. Unlike the untwisted case, the family of SCFTs is in general parametrized, not by M-g,M-n, but by a branched cover thereof. The classification of these SCFTs is carried out explicitly in the case of the D-4 theory, in terms of three-punctured spheres and cylinders, and we provide tables of properties of twisted punctures for the D-5 and D-6 theories. We find realizations of Spin(8) and Spin(7) gauge theories with matter in all combinations of vector and spinor representations with vanishing beta-function, as well as Sp(3) gauge theories with matter in the 3-index traceless antisymmetric representation.National Science FoundationSimons FoundationINCT-MatematicaCoordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Univ Estadual Paulista, ICTP South Amer Inst Fundamental Res, Instituto de Física Teórica (IFT), BR-01140070 Sao Paulo, SP, BrazilUniv Texas Austin, Dept Phys, Theory Grp &Texas Cosmol Ctr, Austin, TX 78712 USAUniv Estadual Paulista, ICTP South Amer Inst Fundamental Res, Instituto de Física Teórica (IFT), BR-01140070 Sao Paulo, SP, BrazilNational Science Foundation: PHY-1066293National Science Foundation: PHY-0969020National Science Foundation: PHY-1316033