Since there is so much confusion about the actual subject of this film I'm going to delve into more detail than I normally do, while attempting to avoid genuine spoilers. While dopplegangers are a running theme, this film metaphorically, indirectly and DIRECTLY addresses the bitter cold war of the times between Communism and Capitalistic ideologies, and the justified fear surrounding the tension between the two--and that of their respective world powers--and how these fears were reflected in Hitchcock's films. This isn't subtle, the idea is repeated over and over, so I'm surprised that so many viewers miss it. It's not that the film really requires deep thought, but it does require paying attention and not being thrown off by the unusual presentation. The narrative, the story is there and completely coherent--especially if you know your history. I've no doubt that Hitchcock would have approved.
This tension and fear (C vs.C, the ongoing tension, and threat of nuclear war) was reflected directly in MANY of Hitchcock's films--such as "The Man Who Knew too Much" and the specifically mentioned "Topaz" and less obviously in many of his other films. "The Birds" wasn't just a fear of nature turning on man (though there is that), but a subtext (or hinted at later in this film by Hitchcock himself) that the Communists would invade or attack and annihilate the innocent (the "duck and cover" scene in the school of "The Birds" was one of the most direct allusions to this paranoia).
Hitchcock's doppelganger story was this film's statement to the effect that the USSR and USA were actually two sides of the same coin and destined to kill each other--as blatantly stated when the narrator asserts that when you meet your double, "one must kill the other" (and repeated every time the film shows archival footage of a U.S.president meeting with one of the Russian leaders--you can't get more obvious than that. The narrator even states this, while reading back excerpts from Hitchcock's diaries.
The genius of the film is also reflected in the juxtaposition of inane TV commercials-designed to support the "American Way of Life" by subverting the public's genuine fears into worrying about competing with the Jones' next door (also financially supporting Capitalism) and substituting the terrifying and all-too realistic terror of all-out nuclear war and global annihilation--to that of pleasing one's husband with a decent cup of coffee. I for one, thoroughly enjoyed the messages of this film and how they were presented--especially the bickering between Khrushchev and Nixon.
FYI: The "Falling man" of this film that some reviewers derided was a real doppelgangerto the famous "falling man" of 9/11, and also a result of the very REAL crash of a small bomber aircraft into the Empire State Building (1945) on a very foggy day (so perhaps that inclusion makes sense now?). A man named Paul Dearing--just as his doppelganger of 2011, jumped to escape the intense flames. This inclusion isn't just cinematic hyperbole or a cheap trick but consistent with the theme of the doomed dopplegangers--with the added caveat: 'Those who can not remember the past are condemned to repeat it.'