How does Gittes not realize he was had by someone? At 4:37 he meets Mrs. Mulwray in his office.
She hires him to spy on her husband Hollis, who she believes in cheating on her. Gittes tails Hollis and photos him in with a young woman. The story, along with the photos, are published in the L.A. newspaper. The next scene is Gittes meeting the real Evelyn Mulwray in his office (20:02).
Would you mistake those two? I wouldn’t. The real Mrs. Mulwray, Faye Dunaway, did not hire Gittes, as the newspaper also mentioned, and is with her lawyer. They serve him with papers in a lawsuit. Gittes is confronted by the real Mrs. Mulwray, "Have we ever met?" "No." "Did I hire you to spy on my husband?" "No." Gittes never explains that he was hired by someone who told him she was Mrs. Mulwray. I’m sure I would have mentioned that.