Movie fans wanting to sample the halcyon days of the 1960s — or, rather, Hollywood’s version of it — are recommended to Kino Lorber’s newly remastered Blu-Ray of “The President’s Analyst” (1967). Never heard of it, you say? There’s a reason for that: J. Edgar Hoover, then head of the FBI, was irked by the film’s depiction of the agency as a cadre of by-the-book buffoons. Calls were made to the White House, which made calls to Paramount Pictures, which pulled the movie from distribution.

Or so the story goes. The FBI did, apparently, visit the set, requesting that producer Robert Evans nix the picture. He rebuffed the agency’s suggestion, only to have the honchos at Paramount Studios twist his arm to the extent that changes were made. As the introductory title card tells us: “This film has not been made with the consent or cooperation of the Federal Board of Regulations (FBR) … or the Central Enquiries Agency (CEA).”

So, yes, mentions of the FBI and the CIA were altered by dubbing their acronyms in post-production — which seems altogether too neat a coincidence for a film whose subject is political paranoia. 

Stranger things have happened, certainly, and if “The President’s Analyst” is not the find its admirers claim it to be, the picture does qualify as some kind of curiosity. Think of it as “Dr. Strangelove” (1964) leavened by “Get Smart” (1965-70) and then put into motion by a creative team huffing-and-puffing to keep up with the zeitgeist. It’s a desperate venture that manages to get by through sheer dint of effort.

Which is another way of saying that if your idea of a good time is watching James Coburn — yes, he of the vulpine smile, measured gaze, and angled physique — don a Beatles wig and groove to some acid-fueled freakadelia, well, then, “The President’s Analyst” comes recommended. 

Coburn is game throughout and the story lets him undercut his macho image. Among the signal moments is when his character, the pacifist Dr. Sidney Schaefer, takes up arms and emerges from an operatic pink cloud in a hail of bullets. “It’s vital,” he tells his comrades, “that we make the public hate The Phone Company.” 

Substitute “Google,” “Amazon,” or “Baidu” for “The Phone Company” and you’ve got yourself a prescient movie, one for whom the Soviets, China, and Canada — yes, Canada — are less worrisome than the machinations of big tech. Thanks to “the miracle in modern communications,” Arlington Hewes (a scarily benign Pat Harrington, Jr.) opines, every individual will have a small chip dubbed The Cerebrum Communicator embedded within his or her brain. All of which is explained by a cartoon graphic that puts a happy face on the threat to individual autonomy.

As I say, prescient stuff, but it takes wading through some clumsy satire to get there. The basic premise of the film is that Dr. Schaefer, being privy to top-secret information as the analyst-on-call for (we assume) Lyndon Baines Johnson, is deemed a security risk — especially when he slips out from under the clutches of his keepers. Whereupon every government in the world wants to kidnap him for the intel he possesses. The only way in which Coburn’s character avoids detection is to join up with a band of hippies making their way across the country. Their van decor, fashion sense, and slang are as groovy as you might fear.

Screenwriter and director Ted Flicker didn’t make many films. He was a TV veteran who scored big in the mid-1970s as co-creator of the hit series “Barney Miller.” Prior to that Flicker was an active hand in the burgeoning improvisational comedy scene, working with talents like Gene Hackman, George Segal, Buck Henry, and Godfrey Cambridge. The latter figures prominently in “The President’s Analyst,” and Cambridge has a grand old time shticking it up with Severn Darden as a pair of competing spies. They’re almost worth the price of admission for this notable, if decidedly ungainly, comedy.