Yep, I've dealt with the same thing on a number of occasions. Luckily, they've recently changed the ordinance enough to make it harder to take my cars, but still fools people of lesser intellect into towing off their total junkers. It used to be, once it was tagged you had anywhere between 3-30 days (depending on the type of infraction and location) to move it or make it legal and roadworthy, or the city would tow it in. If it sat at the impound yard for 90 days it is sold at auction. Now, if it's on private property, they actually have to get a warrant to take the car, but they still stick the same tag to the car, with a letter that warns you that you are in violation of the ordinance. The only give away was that it didn't have a box checked to tell you how long you have to move it. When I called to find out, they informed me that it's just a warning, and if I did nothing then the cop could choose to take pictures and submit them to a judge to order a warrant to get the cars, and they have to tag the car again with the copy of the warrant, and give you 30 days. So I just rearranged my cars (they ran, just not plated) and the neighbor had the junk Corsica with a blown head gasket in his driveway hauled off for scrap, because "At least this way a I get a few bucks for the car. The city is going to come and get it if it's here for another week.". I didn't tell him that that wasn't really the case, I was glad to see that piece of crap gone.
Moral of the story: Always check the letter of the law. Ordinances change all the time, and you might find a loophole.