  We study the electronic structure of chiral and achiral graphene nanoribbons
with symmetric edges, including curvature and spin-orbit effects. Curved
ribbons show spin-split bands, whereas flat ribbons present spin-degenerate
bands. We show that this effect is due to the breaking of spatial inversion
symmetry in curved graphene nanoribbons, while flat ribbons with symmetric
edges possess an inversion center, regardless of their having chiral or achiral
edges. We find an enhanced edge-edge coupling and a substantial gap in narrow
chiral nanoribbons, which is not present in zigzag ribbons of similar width. We
attribute these size effects to the mixing of the sublattices imposed by the
edge geometry, yielding a distinct behavior of chiral ribbons from those with
pure zigzag edges.
