Say you start off floating in space, in a fixed position and orientation, with zero linear and angular velocity, with no external forces. So you are a closed mechanical system. By twisting your body around,
you can't change your linear momentum.
you can't change your position (center of mass).
you can't change your angular momentum.
you can change your orientation (i.e. rotation)!
The fact that you can change your orientation comes as a surprise to me-- why isn't it conserved like the other three quantities? It's a familiar fact-- cats do it all the time in order to land on their feet, and you can find videos of astronauts doing it on the international space station. See the videos linked from https://space.stackexchange.com/questions/2954/how-do-astronauts-turn-in-space . But it still seems counterintuitive to me that they can do this while not being able to change the other three quantities. Is there some intuitively clear explanation as to why?