In this paper, we study zigzag graphene nanoribbons with edges reconstructed with Stone-Wales defects, by means of an empirical (first-neighbor) tight-binding method, with parameters determined by ab initio calculations of very narrow ribbons. We explore the characteristics of the electronic band structure with a focus on the nature of edge states. Edge reconstruction allows the appearance of a new type of edge states. They are dispersive, with nonzero amplitudes in both sublattices; furthermore, the amplitudes have two components that decrease with different decay lengths with the distance from the edge; at the Dirac points one of these lengths diverges, whereas the other remains finite, of the order of the lattice parameter. We trace this curious effect to the doubling of the unit cell along the edge, brought about by the edge reconstruction. In the presence of a magnetic field, the zero-energy Landau level is no longer degenerate with edge states as in the case of the pristine zigzag ribbon.