We give examples of purely infinite simple separable nuclear C*-algebras equipped with actions of the integers and the circle group which are not equivariantly isomorphic to the corresponding actions on the opposite algebras.

To put our result in context: Examples are known of type II and type III factors not isomorphic to their opposite algebras, of separable unital type I C*-algebras not isomorphic to their opposite algebras, and of separable simple unital Z-stable exact C*-algebras not isomorphic to their opposite algebras. No example is known of a simple nuclear C*-algebra not isomorphic to its opposite algebra.

This is joint work with Marius Dadarlat and Ilan Hirshberg.