  Especially in one dimension, models with discrete and continuous symmetries
display different physical properties, starting from the existence of
long-range order. In this work, we that, by adding topological frustration, an
antiferromagnetic $XYZ$ spin chain, characterized by a discrete local symmetry,
develops a region in parameter space which mimics the features of models with
continuous symmetries. For instance, frustration closes the mass gap and we
describe a continuous crossover between ground states with different quantum
numbers, a finite (Fermi) momentum for low energy states, and the disappearance
of the finite order parameter. Moreover, we observe non-trivial ground state
degeneracies, non-vanishing chirality and a singular foliation of the ground
state fidelity . Across the boundary between this chiral region and the rest of
the phase diagram any discontinuity in the energy derivatives vanishes in the
thermodynamic limit.
