Does anyone have any tips on how to ride thru a spin? Not the kind where you expect it and ask them to, the kind where they get mad and surprise you?

Rode a 5-6 yr old mare last night that is normally very well behaved. I hadn't even been on her 30 seconds and she balked. I gave her a bit of leg(no spur), which she ignored, then gave a little more and she started spinning! Needless to say I could only stick with a few spins and bailed. Knocked the wind out of me and I got up spitting dirt but nothing serious.

I can't see the problem, but I think the point was too much hand. And because of that your horse just decided to get rid of you. Anyway, if everything was good before that, try to ride her again, ask someone to watch your riding. If it occures again, ask someone more experienced to ride your mare. As you write that there was very little leg, I nearly sure that the problem is with hands, or bit or teeth. You also can ask vet to check teth, usually they are the source of bad behaviour. Before inviting vet, check if she eats well or shows discomfort? If it's so, the problem is with teeth.
So, good luck. Write again if the problem stays.

She has had her teeth floated withing the last 2 months. She has a history of back trouble, so I told her owner she might have someone come look at her. This is WAY out of character for her.

I don't think it was my hands, I've actually been complimented on how soft my hands are with them. The mare is slowly being brought back into work since her colt has been weaned, but I've ridden her lightly several times and she hasn't been like this.

There were 2 people watching, and another riding in the arena with me. Both of the spectators said there was no warning,she balked and just started spinning. I was only asking for a walk so had a loose rein.
It was partially my fault for not anticipating it, but shes always been so good! I've been riding her off and on for almost 3 yrs and the most shes ever done is balk.

Well, my horse usually balks and if I insist on moving, he raises into gallop and then suddenly stops. I know it's because he tries to get rid of me. He uses this way because I tend to lean forward, so it's his response to that. I saw other people riding and he usually does exactly the same, sometimes qicks but never anything else. It's his way to protest against rider's actions. So, I think it's the same with you and your mare. For her it's just the simpliest way to get rid of you. Why did she dicided to do this is the main question. Maybe she feels pain somewhere, maybe you do something wrong, maybe she is just lazy and doesn't work at all. Anyway, it's better to check her, just to be sure she hasn't health problems, then to check yourself (but as you write, the problem is not with you), then if all thees is ok (i.e. she's in good health, you ride perfectly), maybe to ask someone more experienced to ride her for a while. If it's only behavioral problem, then you'll see that.
As for keeping balance in such spins (I ride English style, so cannot advise properly) but I think you should be relaxed whatever goes wrong (I rode spooky horses and they used to make different jumps in different sides under the saddle. Only relaxed back and pressed knees saved me).

Lol, it's entirely possible that something I did just pushed her over the edge and she threw her fit, but I didn't do anything different than I've done w/her in the past. I just don't understand why she would react that way when I've pushed her much harder in the past and she was fine. If there is nothing physically wrong, I think she just wanted to get rid of me. Shes a very "bossy" mare,lol.

You say she has a history of back trouble, I would definitely be thinking about that one... I've had the same circumstances with a horse having back out of whack/issues/pain.

I hope you figure it out!