Rock-paper-scissors?

With more class unlockable updates coming (eventually), I've seen a lot of people talk about the "rock-paper-scissors" relationships of some classes.  The idea is that for every class, there's another that it tends to beat easily, and yet another that it tends to get pwned by (i.e. Pyro kills Spy and is killed by Soldier).

My question is, can we come up with a rock-paper-scissors for the entire class roster?  I started one, and I like most of it, but I got stuck with a soldier/heavy pairing that I'm not sure about.  Also, I couldn't fit the Medic in anywhere.  Here's where I ended up:

Scout:  kills Demo, killed by Engi
Soldier:  kills Pyro, killed by Heavy (???)
Pyro:  kills Spy, killed by Soldier
Demo:  kills Engi, killed by Scout
Heavy:  kills Solder (???), killed by Sniper
Engi:  kills Scout, killed by Demo
Medic:  uh... Buttslaughter?
Sniper:  kills Heavy, killed by Spy
Spy:  kills Sniper, killed by Pyro

This was just my impression.  Are there other combinations that work better?

(EDIT: Oops, goes in General TF2 Discussion.)

I think it's more complex than that - I'd go for the following:

Scout: Kills demoman, medic, sniper; killed by HW, engineer
Soldier: Kills pyro, medic, soldier; killed by HW, engineer, soldier
Pyro: Kills medic, spy, sniper, hurts scout; killed by engineer, solider, HW
Demo: Kills engineer, HW; killed by pyro, scout
Heavy: Kills scout, soldier, pyro, heavy, medic; killed by demo, sniper, spy
Engineer: Kills scout, soldier, pyro; killed by demo, spy
Medic: Kills spy (sometimes); killed by sniper, scout, pyro, soldier, HW
Sniper: Kills HW and sometimes solider; killed by scout, pyro, spy
Spy: Kills Engineer, Sniper, HW; killed by pyro, sometimes medic

Well good except that sniper also kills engineer, scout also kills pyro, and I would argue that medic kills scout.

skill overcomes everything, never think about the game this way.  You're just going to get into a fight thinking your going to lose if you start to believe this crap.

Yep.

Yeah its true skill can overcome anything. Sure "essentially" a Scout can pwn a Demoman, but a skilled Demoman has his/her timing down and can shoot a grenade right into a Scouts face. Just because it looks improbable doesn't make it impossible.

Every class has weaknesses.  There are some classes that can exploit certain weaknesses better than others.

It's not so much about A beats B, but rather that A and B working together make up for each others' weaknesses.

'swhy it's called TEAM fortress, ya?