  The two-flavor Gross-Neveu model with U(2)$_L\times$U(2)$_R$ chiral symmetry
in 1+1 dimensions is used to construct a novel variant of four-fermion theories
with O(2)$_L\times$O(2)$_R$ chiral symmetry. The spontaneous breaking of the
group O(2), a continuous group with two connected components (rotations and
reflections), gives rise to new phenomena. It is ideally suited to describe a
situation where two distinct kinds of condensation compete, in particular
chiral symmetry breaking (particle-hole condensation) and Cooper pairing
(particle-particle condensation). After solving the O(2) chiral Gross-Neveu
model in detail, we demonstrate that it is dual to another classically
integrable model due to Zakharov and Mikhailov. The duality enables us to solve
the quantum version of this model in the large $N$ limit with semiclassical
methods, supporting its integrability at the quantum level. The resulting model
is the unique four-fermion theory sharing the full Pauli-G\"ursey symmetry with
free, massless fermions (``perfect Gross-Neveu model") and provides us with a
solvable model for competing chiral and Cooper pair condensates, including
explicit soliton dynamics and the phase diagram.
