On action-angle duality

Action-angle duality is a property enjoyed by systems of Ruijsenaars type - many body systems; relativistic analogues of Calogero-Moser-Sutherland systems - whereby families of integrable systems come in natural pairs: the canonical coordinates of one system are the action-angle variables of the other, and together they generate the whole phase space. I will explain this property, and why it is special. When transported to quantum systems, the action-angle duality property is represented in the form of bispectral operators.
I hope also to describe results obtained with László Fehér in which Hamiltonian reduction is used to obtain systems in action-angle duality relation with one an other.