One morning, Jan Bures (or is it Karel Bures? I forgot which is which :) discovers that his twin brother Karel (Jan?) has choked to death on a bread roll. However, Jan knows what to do - Karel was a pilot for Universum, a time travel agency. So he dresses up as his brother and goes to his brother's job. He soon discovers that his brother was part of a Nazi ploy to hijack a time travel rocket, go to 1944 (when Germany is in trouble) and give Adolf Hitler an A-bomb. Although he can do little to prevent this, the ploy fails - firstly, the hijackers are double-booked with two American tourists, and secondly, they land in 1941 - when German soldiers are threatening to conquer Moscow. When they return, they return before they actually took off (this is Jan's attempt to save his brother by preventing him from suffocating). And things go downhill from there...
Firstly, this is a very original take on the topic of time travel. Secondly, Petr Kostka does a great job in this double-role (which is in fact more of a single role after all). The effects and styling appear naive now, but they're good for their time and place. (The Universum scenes were mostly filmed in the then-new Prague subway.) Finally, the writers and director must be commended for not getting lost in the screenplay and for not letting us get lost or bored - there's always something going on and if you're willing to believe that time travel was possible in the 1970s as demonstrated in the film, there are few (if any) plot holes or inconsistencies. Music doesn't play much of a role here.
The near-obligatory compromises to Communism (this was the 1970s, y'know) are present, but in a way that just makes it even more entertaining - the aforementioned American tourists react to the sight of Hitler in the flesh by demanding to take photographs with him, much to the Führer's frustration (the guy who played him was great too). Can't really think of any other examples right now.
If you like this film, an obvious recommendation would be Zabil jsem Einsteina, pánové (I Killed Einstein, Gentlemen), but I found Zítra vstanu a oparím se cajem more entertaining.