  We address the interplay between local and global symmetries in determining
the continuum limit of two-dimensional lattice scalar theories characterized by
$SO(N_c)$ gauge symmetry and non-Abelian $O(N_f)$ global invariance. We argue
that, when a quartic interaction is present, the continuum limit of these model
corresponds in some cases to the gauged non-linear $\sigma$ model field theory
associated with the real Grassmannian manifold $SO(N_f)/(SO(N_c)\times
SO(N_f-N_c)$), which is characterized by the invariance under the color-flavor
reflection $N_c\leftrightarrow N_f-N_c$. Monte Carlo simulations and
Finite-Size Scaling analyses, performed for $N_f=7$ and several values of
$N_c$, confirm the emergence of the color-flavor reflection symmetry in the
scaling limit, and support the identification of the continuum limit.
