Think of the reins as baby birds; you don't want to hold it too tight, but you don't want to let them fly away. And make sure you keep your elbows bent and your thumbs up, that should help with keeping your hands more symmetric.
Try to relax your shoulders, it's linked. And try to follow your horse's movements :)
You can attach strings with the breastplastes and when you hand your reines you also hand the string it helps you to keep a good position and the fixed hands during riding :)