Jax DOES spin at the end of his counterstrike.

Akali and Amumu*, and Xin Zhao* added.

Fizz, Janna, and Riven added as honorable mention.

Ahri, Dr. Mundo, Kayle, Nocturne, Orianna*, and Warwick don't spin in their spell animations, they just turn a bit.

Vayne rolls as movement, but it doesn't cause damage directly.

Roundhouse champions don't qualify.

Nidalee I haven't been able to confirm yet, holding off.

* a few champions it's hard to tell whether they're actually spinning their body 360 degrees or not due to a fast animation speed

Actually every single one of those fully rotates 360 degrees (except Mundo, just his fire and cleavers) during one of their spells (admittedly Kayle's is her heal spell). Just saying, that was the requirement.

That's a full 360 degree turn, and just as a point of comparison, Renekton and Darius operate on the same concept except with a weapon.
Riven spins. Look at Broken Wings. First to the left, then to the right, and then a flip.

Kog'Maw spins when he is chasing his tail after idle for a couple of seconds... Kog'Maw is an insane lategame champion... CORRELATION????????

Lee Sin I re-watched and I was mistaken, Evelynn I have been unable to find a decent video so far so I'm not sure, I'll hold off for now.

Orianna I may be incorrect about as it's hard to tell with the speed of her animation, the rest clearly just turn partway then turn back, unless they have multiple animations or their animation has been changed so that they do actually rotate 360 degrees.

As for Kog'maw idle animations don't count, just ability animations.

Ahri: Look at 0:25 http://www.youtube.com/watch?v=hlalkAUkwqE
Nocturne: Look at 0:27 http://www.youtube.com/watch?v=owrwt97AH54
Riven: From 0:27-0:32 http://www.youtube.com/watch?v=V_ERvayBr3M

Would post others, but Kayle's is just for a heal animation, Mundo doesn't spin himself, and I didn't find a good video for Warwick or Evelynn yet.

Ahri and Nocturne spin on next ability and attack respectively when their passive procs so it's more of a modifier so I didn't include them, plus Ahri just sends out a skillshot anyway. Nocturne actually gets his off quite often in a fight though and it is a AoE hit around him so I have decided to include him after all. Riven's spin was so fast I completely missed it when I checked before, but I slowed the video down and you were right.

Sorry, but, most of these guys don't spin. Except for Ahri during her ult and POSSIBLY Lee during his. Amumu doesn't spin either, he just whips his head around in a circle. Nocturne turns totally sideways and does an odd wiggle, but to my knowledge does not spin, MAYBE during his shield. WW doesn't spin unless he uses it as harass, and then that's just the player making him walk in a loop. Mundo does not spin. Orianna MIIGHT spin, but I've played her quite a bit and I never noticed it, she definitely DOES spin a lot during her dance though.... time to go play some chaps I haven't in a while... Oh, and Nidalee DEFINITELY DOES NOT SPIN AT ALL. Neither does Akali, ever.

Edit: Tested and Riven does 2 360 spins on the Y axis and one on the X axis in the course of her full Q combo. Orianna spins when she shields. I never noticed it because I usually have my eyes fixed on someone else's attack animation when I shield.

Spectated some chmapions in slowmo.
Nidalee spins.
Evelynn spins near as I can tell.
Orianna doesn't spin, she does a 180 and goes back the way she came.
Amumu spins.

Ahri *does* spin on her W. I'm pretty sure she also spins on her ult sometimes, but it's more of a barrel roll/corkscrew thing rather than the renowned Garen spin.

Edit: Also, Nidalee DEFINITELY spins. It's part of her Q animation in cougar form.

Ahri does during Foxfire, as I said. And her ultimate does include a barrel roll, which is the third possible axis for a spin (one horizontal, two verticle axes).
Lee does spin during his ultimate, so does Evelynn during Ravage.
I did mention that I could be wrong about Amumu.
Watch the video I linked, Nocturne does a full spin.
Warwick spins with his Q.
I did say Mundo doesn't spin, just his fire and cleavers.
Orianna spins when casting spells.
And Nidalee's cougar form has a spin attack, and Akali's E is a spin attack.

Spin2win srsbsns