  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,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.
