Even simpler, activate the Clipping. Everything that you push toward a plane of symmetry will be flattened onto it and stick to it. Once a vertex is at the middle, you can’t move it away any more but you can still move it in all other directions.
No need to worry about the 3D cursor, you can use any pivot you want… And you don’t need a hotkey.
Sorry guys, I completely forgot that I posted this question. Thank you very much for the help. I think Kaluura’s solution is fantastic here, Clipping! I’m going to have that on most of the time now, thanks!