Charles Hope in his book "A Horseman's Manual" claims that early Assyrian horsemen rode near the rear, a claim that is repeated on wikipedia as "However, these riders sat far back on their horses, a precarious position for moving quickly, and the horses were held by a handler on the ground, keeping the archer free to use the bow. Thus, these archers were more a type of mounted infantry than true cavalry"
Were there any major differences, beyond the lack of stirrups, in ancient horseback riding?
You could check out the info in "The Horse, The Wheel, and Language" about early horse riding. Apparently if you ride a donkey, which I think the Assyrians had before they had horses, you do need to sit further to the rear than if you ride a horse.