That sounds exactly the problem. Everytime you wrap a coil, you'll put a twist in the rope. It's most noticeable on electrical cabels.
I've been coiling ropes for most of my life (tree surgeon). I put a reverse twist on every gather of rope as I go, elbow to hand is the worst, so just gather to one hand, reverse twisting and adjusting loop length.
There is a very easy way to coil rope or any cordage to remove the twist - no wrist action (snigger!) needed. You essentially coil in a figure of 8 between your hand and neck. Very difficult to explain in words, but it enables me to coil a 60m 10mm climbing rope in about 90 seconds.
1) Find the middle of the rope (not essential, but makes it quicker, and reduces the likelihood for knots/tangles
2) hook the middle of the rope around your thumb, and stretch out your arm to the full extent, holding it away from your body
3)with your other hand, pass the (doubled) rope behind your neck, then hook it back onto your thumb.
4) repeat, but each time you pass behind your neck, alternate the shoulder you pass it behind, thus the rope crosses over itself each pass, and it forms a figure of 8 between your thumb/hand, and your neck (it is this action that magically removes the twist)
5) when you have a couple of meters of rope tails left, lift your hand/thumb that is holding the rope, and shrug it off your head- you will end up holding a set of uniform non twisted loops of rope in your hand
6) with your spare hand, wrap the tails of the rope around the hank 3 or 4 times, and tuck in on itself to secure (or if you want to be really clever, there is a better way to finish this off that uses no knots, and turns the tails of the rope into two handy carrying straps for the rope - but having just attempted to describe that in full, i have realised its impossible in words and deleted it!)