Although I have seen many perfectly symmetric nozzles, I have also seen designs such as this (Virgin Galactic) which is only symmetrical about one axis:
Why is this?
Space Exploration Stack Exchange is a question and answer site for spacecraft operators, scientists, engineers, and enthusiasts. It only takes a minute to sign up.Sign up to join this community
According to comments on this thread, the shape of the nozzle on SpaceShip Two is to provide "free pitch-up" during the short powered flight period. That nozzle shape would direct thrust force downward slightly, under the craft's center of gravity, causing it to slowly rotate nose-upward. Presumably modifying the nozzle shape was easier than tilting the whole engine, and the nozzle can be customized for different flight profiles (i.e. more or less fuel carried on a particular flight).
In this video, you can see the pitch-up maneuver during the powered burn from roughly 0:30 to 1:00. Ordinary aircraft achieve pitch-up by using control surfaces (elevators/elevons), but using these creates drag. Directing the rocket motor thrust instead gives a pitching force, with less drag.