Twisted-pair wire's effect on Common mode noise
It's actually magnetic pickup.

If you go all the way back to laws of physics, electromagnetic fields passing through a loop of conductor generate currents, and its going to increase with the area of the loop.  Twisted pairs minimize the area of the loop, and signals generated in successive twists tend to cancel.  (Horowitz and Hill, 2nd edition, page 456).

That's probably the main effect.  I like to think that much of remaining EMI on each lead of the twisted pairs will be similar (and thus common mode), as the close proximity of the wires exposes them to very similar EMI profiles.  That's probably not as big an effect as just minimizing differential EMI induced noise, as above.

Practically speaking, an old trick to make twisted pairs out of plain old stranded hookup wire is to run two lengths of wire next to each other, tie one end down, chuck up the free end in a power drill, and use the drill to wind the pair.  Go a little more than you think you should, as it will open up a bit on release.