  We rewrite the two-dimensional SU(2)$\times$ SU(2) chiral spin model in terms
of SO(3) and {\bf Z}$_2$ degrees of freedom. The transformation, which is
motivated by a similar representation of the corresponding lattice gauge theory
in higher dimensions, exhibits the presence of dynamical SO(3) vortices and
associated strings. We present arguments that (pairs of) SO(3) vortices with
long strings play a crucial role in disordering the spin system at arbitrarily
low temperatures.
