  We exhibit examples of simple separable nuclear C*-algebras, along with
actions of the circle group and outer actions of the integers, which are not
equivariantly isomorphic to their opposite algebras. In fact, the fixed point
subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit
are well behaved from the perspective of structure and classification of
nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite
nuclear dimension. One is an AH-algebra with unique tracial state and absorbs
the CAR algebra tensorially. The other is a Kirchberg algebra.
