Polyelectrolyte-surfactant complexes (PE-S) formed by double hydrophilic cationic polyelectrolyte poly[3,5-bis(dimethylaminomethyl)-4-hydroxystyrene]-block-poly(ethylene oxide) (NPHOS-PEO) and anionic surfactant sodium dodecyl sulfate (SDS) in acidic aqueous solutions were studied by light scattering, SAXS, and scanning transmission electron microcopy in the environmental mode (wet-STEM) for various stoichiometric ratios between the numbers of SDS anions and dimethylaminomethyl groups of NPHOS in the complex. The obtained results show that the NPHOS-PEO/SDS system behaves differently from other systems of double hydrophilic block polyelectrolyte and oppositely charged ionic surfactant because it forms water-insoluble PE-S for compositions close to the zero net charge of the complex. This phase separation occurs, instead of the PE-S rearrangement to core-shell particles, which is hindered due to conformational rigidity of the NPHOS blocks. For the surfactant amounts below and above the precipitation region, large spherical aggregates and their clusters are present in the solution. SAXS measurements indicate that although the NPHOS-PEO/SDS system does not form the core-shell particles with the NPHOS/SDS core and the PEO shell as other PE-S of double hydrophilic polyelectrolytes, the aggregates contain domains of closely packed surfactant micelles which bind to both NPHOS polyelectrolyte blocks and PEO blocks.