Pretty sure I remember it being possible to do a 4 dimensional rotation that's equivalent to a 3D mirror image. I'm at work now though and can't do the math for a while
Everything gets easier when you add dimensions - just ask the string theorists.
Sure. Add an extra dimension with coordinate w to 3d space.
Then just rotate around the (x,y)-plane until z -> -z and w -> -w. Everything in the (x,y)-plane stays the same and z has been inverted.