I'm studying Bondi's work on gravitational radiation. He applied axial symmetry and reflection symmetry to a source of gravity. How is it that axial symmetry does not automatically imply reflection symmetry? The assumption for axial symmetry is [itex]\phi \rightarrow \phi'= \phi +[/itex] constant The assumption for reflection symmetry is [itex]\phi \rightarrow \phi'= -\phi[/itex] The first condition implies the second if the constant is [itex]-2\phi[/itex], so I don't see how the second assumption adds anything new to the source. Also, I read that the second assumption prohibits the solution from rotating, but I don't see why this distinction needs to be made since it is invariant under any rotation.