The problem: Horse starts a spin with head low but raises it in the spin.
Why does this problem happen? A horse raises his head during a spin because he gets out of alignment or he just wants to leave!
How to correct this problem: If my horse is just learning to spin and raises his head in the spin, I might walk him out of it, correct any body alignment problems, ask him to lower his head and neck, ask for a spin again and repeat as necessary. If he is trained and raises his head in the spin, I will ask him to lower it again while he is in the spin (assuming his body position is correct), then finish the spin with his head low. To do this, I use gentle bumps with the inside rein if I cannot see his inside eye or the outside rein if he is overbent. Consistency is everything here - if I ask him to lower his head every time he brings it up, he will eventually see the futility of doing so.
Note: It is extremely important to end the spin with the horse’s head low. Otherwise he may think putting his head up is the way to get rewarded! If I absolutely cannot convince him to lower his head in the spin, I end the spin then ask him to lower his head...