There is a way to coil a cord that doesn't twist it along its length: the over/under technique. I started writing out a long description, but it makes more sense if you can see it done:
(You might want to watch it full screen. Just click on "YouTube" at the bottom of the box above. [You'll have to start the video first.] I didn't know that either until I tried it just now.)
The first time a particular cable is coiled is the hardest because the cable was shipped coiled the "usual" way, which twists the cable along its entire length and makes it misbehave. So you will have to untwist the cable the first few times as you coil it. But once you "train" the cable, it is (almost) a hassle free technique. (I have a hundred-foot power cable for the trimmer. To uncoil it I just lay it on its side, grab the "top" connector, and wander around the yard until it's played out.)