you can put an axis anywhere you want
but when you compare your position to the “position” of the virtual person im the mirror there has been a virtual rotation along the up down axis. of course there was no rotation at all just we pretend there is another person in the mirror and according to his perspective, his right is our left.
forget the mirror, stand facing another person. you have the same perspective as far as up-down. also east west. north south, but right left are called depending on whos perspective. this is just a convention of nomenclature but does not represent any reality