Both textbooks I'm currently reading (Mathematical Methods of Classical Mechanics by Arnol'd - get it here - and Classical Mechanics by Shapiro - here) say that, in the case of the planar harmonic oscillator, using polar or cartesian coordinate systems leads to different action-angle variables (and I'm ok with that) and different invariant tori.
I think I've understood in what sense those tori could be different (i.e. not diffeomorphic at all), but I'd be very glad to see a proof (or even read an insight)