  Recently a paper on the construction of consistent Wigner functions for
cylindrical phase spaces S^1 x R, i.e. for the canonical pair angle and angular
momentum, was presented (arXiv:1601.02520), main properties of those functions
derived, discussed and their usefulness illustrated by examples. The present
paper is a continuation which compares properties of the new Wigner functions
for cylindrical phase spaces with those of the well-known Wigner functions on
planar ones in more detail. Furthermore, the mutual (Weyl) correspondence
between Hilbert space operators and their phase space functions is discussed.
The star product formalism is shown to be completely implementable. In addition
basic dynamical laws for the new Wigner and Moyal functions are derived as
generalized quantum Liouville and energy equations. They are very similar to
those of the planar case, but also show characteristic differences.
