In one scene being shot on the lower west side of New York City, James Coburn was being chased by two uniformed NYPD officers, who were just actors playing New York cops. Shouting "Stop, stop", they chased Coburn around a corner and he ran into a real NYPD officer, who dropped him with one blow from his nightstick. Coburn had to seek medical treatment, which postponed the filming for a time.
A not-so secret fact is that the FBR and CEA were originally identified by their real names; apparently pressure was brought to rename them after the film was shot. Thus every line where FBR or CEA is spoken has been redubbed, often very audibly. If you watch actor's lips, they're really saying FBI and CIA.
CEA agent Don Masters (Godfrey Cambridge) wears a "Dizzy Gillespie for President" sweatshirt at the beginning of the movie. Legendary jazz trumpeter Dizzy Gillespie actually ran a short-lived campaign for the office in 1964 which, while not entirely serious, appealed to many disaffected voters.
The head of the "FBR" is named Lux. "Lux" was a then-popular brand of vacuum cleaner. At the time, the director of the FBI was J. Edgar Hoover (who had the last name of another popular vacuum cleaner)..
According to Larry Karaszewski of the website trailersfromhell.com, writer/director Theodore J. Flicker said the monologue on the "N" word was based on a true story told to him by Diana Sands.