  We study the phase-ordering kinetics following a quench to a final
temperature $T_f$ of the one-dimensional p-state clock model. We show the
existence of a critical value $p_c=4$, where the properties of the dynamics
change. At $T_f=0$, for $p\le p_c$ the dynamics is analogous to that of the
kinetic Ising model, characterized by Brownian motion and annihilation of
interfaces. Dynamical scaling is obeyed with the same dynamical exponents and
scaling functions of the Ising model. For $p>p_c$, instead, the dynamics is
dominated by a texture mechanism analogous to the one-dimensional XY model, and
dynamical scaling is violated. During the phase-ordering process at $T_f>0$,
before equilibration occurs, a cross-over between an early XY-like regime and a
late Ising-like dynamics is observed for $p>p_c$.
