Did you pay with a credit card or check?
I would call a lawyer (even one in your homestate if you want, to get legal advice - and they can then coordinate with someone in Oregon when the times comes). I can recommend you a good lawyer here in Portland for traffic related issues (503-675-4330) ... I don't know if they would handle this kind of case, but at the very least someone should be able to point you in the right direction. Probably not open on Easter Sunday though, I think they have a messaging service. I would assume they can point you to a lawyer that can handle your case -- at least -- if not take it themselves. Assuming your story is 100% legit.
Know this: Under Oregon law, towing companies can sign contracts with property owners for the exclusive right to patrol private lots and haul off improperly parked vehicles. Drivers, typically, are paid on commission -- the more cars they tow, the more money they make.
So, where you were parked matters.
I would stop payment on your check or credit card. If you are telling the truth, and were parked legally, and not cited by the police or in any zone that was marked as "private property" with "will be toed" signs. Get pictures of where you were parked. Stop payment on any payment you made. And contact a lawyer. If you really did not do anything wrong -- you have the legal right here, and a lawyer will probably take your case to get a % of the winnings when they sue the company for illegally towing your car.
The business demanded cash only, which is even more of a red flag. So no - there was no credit card or check involved.