We present studies on the structure of complexes of the cationic, bilayer-forming surfactant, didodecyldimethylammonium bromide (DDAB), and the anionic polyelectrolyte sodium polyacrylate (PAANa). In the presence of uncomplexed polyelectrolyte in the coexisting aqueous solution, these complexes are found to exhibit a swelling transition followed by a deswelling transition on increasing the salt concentration. Lamellar structures with low periodicities occur at both low and high salt concentrations, which are stabilized by polymer bridging and van der Waals attraction, respectively. The swollen complex found at intermediate salt concentrations forms the sponge phase. Our results reveal that polyelectrolyte adsorption on bilayers has a profound effect on inter-bilayer interactions. The polymer-induced interaction changes from being attractive to repulsive as the surface coverage increases on increasing the salt concentration. Our results also confirm that polymer adsorption alters the elastic moduli of the bilayer, in agreement with earlier theoretical predictions.