I'm asking about functional asymmetry - doors on one side but not the other wouldn't count.
I can think of three examples:
- experiments with oblique wing designs, such as NASA's utterly hideous AD-1, designed apparently by Burt Rutan
- displaced fuselage designs, such as the Blohm & Voss BV 141 or (another Burt Rutan aircraft), the Boomerang
- all single-propeller aircraft, in which the torque of the engine exerts asymmetrical forces on the aircraft
Symmetrical asymmetry doesn't count
(Then there are some that I don't think really count - for example, the Wright brothers' Flyer 1 placed the engine and the pilot side-by-side, but since they were intended to balance each other out and the twin propellors were symmetrically placed, that counts as an attempt at symmetry as far as I am concerned!)
Other examples, and the reason for their asymmetrical design
Are there any other good examples? The more ugly and wrong-looking, the better.
I'm interested in the problems that the asymmetry aimed to solve - for example, oblique wing designs aimed to procure aerodynamic advantages (which turned out to be at the expense of handling issues), or Rutan's Boomerang which "was intended to be a multi-engine aircraft that in the event of failure of a single engine would not become dangerously difficult to control due to asymmetric thrust" (Wikipedia)
Just for reference, here's the AD-1 (NASA, via Wikipedia):
... and the BV 141 (Deutsche Bundesarchiv, via Wikipedia):