gameboyguy13 has been a superstar, comforting me in my time of confusion and WTF-ery, and helpfully linking me to California law on the towing of vehicles. They appear to be charging the maximum towing fee, maximum storage fee, and an additional "dolly fee"; apparently my car required special handling of some sort. Since we're to be picking it up within the initial 24 hours, they shan't gouge me for two days of storage. And from reading the law, either the "parked on private property" is entirely trumped-up and bogus, or instead of strictly "parked on private property" it would have instead been impeding access *to* private property, which is likewise bull, but a rather more plausible variety of bull, and they have to have photos to prove this bull if that's what it was.
My aunt is taking me there; her arrival is imminent.