It's not about a swing "slightly different". In real life the right swing portrayed in the game would be weak and slow as fuck, and a mere hit to the weapon would disarm you. (you don't ever cross your arms like that, unless you want to be disarmed) In real life, you would change your grip if you wanted to do a right swing, or you wouldn't do a right swing at all and stick to the other directions of attack.
One thing that could be done if you really wanted to attack from the right without changing your grip would be a hit with the back of the polearm. It would still be somewhat weak, but less, much less goofy looking.
Edit : I don't care about gameplay elements like balance and "OMG POLARM OP PLZ NERF". Balance can be discussed later.
Edit2 : The left swing is the fastest (read the only useful swing) of them all. I remember when i thought that reach was the most important and used only stab and right swing. I got killed all the time because of facehugging 2handers.