with $p=2$. Phase-ordering proceeds by means of formation and subsequent growth of domains through interface diffusion and annihilation. This similarity goes beyond the qualitative level: we find the same exponent and scaling functions for every $p\le p_c$ and for all the one-time or two-time quantities considered. This reflects a deeper similarity than what a unique universality class, involving only the value of the exponents, would imply. For $p>p_c$ the dynamics changes dramatically, due to the relevant role played by textures. While for $p\le p_c$ textures are quickly removed by means of non-activated processes, for $p>p_c$ their removal can only be realized through activated processes. For quenches to $T_f=0$, activated process are forbidden, and, therefore, textures remain in the system up to the longest times. Their peculiar growth mechanisms characterize the dynamics, similarly to what happens in the one-dimensional XY model, with the notable feature of violation of dynamical scaling and the anomalous growth with $z=4$ of the winding length $L_w(t)$. For quenches to finite $T_f$, textures survives up to a characteristic time $\tau _p^{cross}(T_f)$ which can be rather long for small temperatures or large $p$. A crossover phenomenon is then observed from an initial dynamics of the XY type, to 