The symptoms you describe sound like head shaking to me.
Head shaking symptoms are almost as varied as there are horses with Headshaking Syndrome. Below is a list of some
of the symptoms, but not all horses have all of them :
Symptoms start in March/April and end in September/October.
Scratches his nose on his foreleg or an object.
Buries his nose in the tail of the horse in front of him.
Personality changes - an unhappy horse.
Constant sneezing/snorting while working.
Shakes at trot/cantor, but not at a walk.
Suddenly jerks his head up almost hitting the rider.
Constant blowing nose as if to expel something.
Shakes violently enough to lift front legs off of the ground.
Jerks suddenly like a bee just stung his nose.
Kicking one or the other foreleg towards their head.
There are different treatments available, over here in the UK, we have a piece of netting/mesh material which attaches to the noseband and covers the nose. Try doing a search on google for head shaking in horses.
I'm not a vet, so I'm not a 100%, but the symptoms you have said, does sound like it.