We return to our study \cite{BEH} of invariant spin fields and spin tunes for
polarized beams in storage rings but in contrast to the continuous-time
treatment in \cite{BEH}, we now employ a discrete-time formalism, beginning
with the PoincareËŠ maps of the continuous time formalism. We then
substantially extend our toolset and generalize the notions of invariant spin
field and invariant frame field. We revisit some old theorems and prove several
theorems believed to be new. In particular we study two transformation rules,
one of them known and the other new, where the former turns out to be an
SO(3)-gauge transformation rule. We then apply the theory to the dynamics of
spin-1/2 and spin-1 particle bunches and their density matrix functions,
describing semiclassically the particle-spin content of bunches. Our approach
thus unifies the spin-vector dynamics from the T-BMT equation with the
spin-tensor dynamics and other dynamics. This unifying aspect of our approach
relates the examples elegantly and uncovers relations between the various
underlying dynamical systems in a transparent way. As in \cite{BEH}, the
particle motion is integrable but we now allow for nonlinear particle motion on
each torus. Since this work is inspired by notions from the theory of bundles,
we also provide insight into the underlying bundle-theoretic aspects of the
well-established concepts of invariant spin field, spin tune and invariant
frame field. Since we neglect, as is usual, the Stern-Gerlach force, the
underlying principal bundle is of product formso that we can present the theory
in a fashion which does not use bundle theory. Nevertheless we occasionally
mention the bundle-theoretic meaningof our concepts and we also mention the
similarities with the geometrical approach to Yang-Mills Theory