A horse that rears and bucks alternatively is a very hard thing to deal with as the techniques used for each are very different. And you can correct me here Kevin if I don't describe this well.
For a rear you need to be allow room for them to go forward so they don't continue to be 'stuck' in the rear. If they are REALLY going to rear with intention it helps to keep a good bend in the neck, however this should ONLY be done before they rear as a prevention. If they make it up into the air you shouldn't interefere with the balance whatsoever, come forward in the saddle up on to the neck to counter balance the horse. If you have to hold on to something make it the mane or if they really get some height you can hold on to their neck. In this instance, wait till return to earth, give them room to go forward again. This is hard if they are rearing with intent as you have to have very good balance to ensure that you can balance independantly of the horse, similar to jumping. This is so hard because the timing has to be just right to a.) avoid the causing the horse to fall b.) encourage forwards movement and c.) discourage further rearing.
Now a buck is almost the opposite. When a horse bucks they CAN continue in a forwards direction and will soon discover that bucking is the hardest way of doing this if you make this very clear to them. Also completely different is your balance - you want your feet forward, your heels down and to lean back slightly. Keep the horses head from getting between their legs so they can't actually throw in something pro-rodeo style. Personally for a bucker I keep them going and give them a good smack with a crop for each buck they give me until they give up. The smack is ALWAYS a decent one (no half-arsed 'taps') and ALWAYS behind the saddle (so they keep going forwards). Hence my comment 'one hand for reins, one hand for crop...' Now this works for me, usually after the first two or three bucks they give up. As I said, anyone can choose to offer alternative advice.
The problem with this situation is that if a horse is going from rearing to bucking, it takes someone with a hell of a lot of experience to deal with this effectively. Rear - riders weight is forward, not interefering with horses balance to buck - riders weight is back, keeping the head from getting too low. It is exceedingly hard when these maneuvres are alternated in quick succession. Luckily there are very very few horses that are capable of pulling such a stunt. If this is indeed how the horse in question is behaving, I would not be ashamed to leave it to a professional, it's just not worth getting hurt trying to someone a favor. With horses like this, it isn't enough just to 'stay on' you also have to ride effectively so that you can train them of this, not a small task at all.
Good luck with whatever you decide to do and stay safe!
All horses deserve, at least once in their lives, to be loved by a little girl.