A horse that behaves this way under saddle, whilst having a canter is a form of expression that he is enjoying himself.
Excitement
Some get excited within the company of other horses and will mess around at the beginning of a ride then settle.
Fresh
Keeping a horse in a stable for long periods of time will leave him with a lot of pent up energy to release when ridden.
It would be sensible to turn him out or lunge him to release some energy before hand.
Fit
Working horses that are fit and stabled might feel like a coiled spring once on the move.
Lunging before the rider gets aboard will help take the explosive edge off him.
Pain
Make sure the saddle is fitting correctly.
A saddler can check this over for you, if your not sure.
An ill-fitting saddle can cause pain and discomfort.
Regularly check your horse's back for any rubbed hair or sore patches.
A saddle may need re fitting if he changes shape due to weight loss or gain or changes in fitness level.
Girth
Make sure the girth is not pinching the skin.
The hair should be lying flat behind the girth.
To iron out the creases, lift up a front leg and pull it forward, following with the other leg or lead the horse round in a tight circle in both directions to iron out the creases.
This will also give the saddle time to warm up before you mount the horse.
Irritation of any kind like an insect bite which is usually on the hind quarters can trigger a reaction from the horse.
Teeth
Again if the horse is in pain this could be his way of reacting.
A horse may give one buck the first few times they have a saddle on their back, if they don’t have proper ground training.