  We consider a class of discrete $q$-state spin models defined in terms of a
translation-invariant quasilocal specification with discrete clock-rotation
invariance which have extremal Gibbs measures $\mu'_{\varphi}$ labeled by the
uncountably many values of $\varphi$ in the one-dimensional sphere (introduced
by van Enter, Opoku, K\"{u}lske [J. Phys. A 44 (2011) 475002, 11]). In the
present paper we construct an associated Markov jump process with quasilocal
rates whose semigroup $(S_t)_{t\geq0}$ acts by a continuous rotation
$S_t(\mu'_{\varphi})=\mu'_{\varphi +t}$. As a consequence our construction
provides examples of interacting particle systems with unique
translation-invariant invariant measure, which is not long-time limit of all
starting measures, answering an old question (compare Liggett [Interacting
Particle Systems (1985) Springer], question four, Chapter one). The
construction of this particle system is inspired by recent conjectures of Maes
and Shlosman about the intermediate temperature regime of the nearest-neighbor
clock model. We define our generator of the interacting particle system as a
(noncommuting) sum of the rotation part and a Glauber part. Technically the
paper rests on the control of the spread of weak nonlocalities and relative
entropy-methods, both in equilibrium and dynamically, based on
Dobrushin-uniqueness bounds for conditional measures.
