It looks like a fraud scheme that dealers use . It's called the yo yo scheme.
When parties rescind the contract that means that both parties must be put in the same position they were prior to the contract.
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You're right, dealer is wrong. I've blogged about this Yo Yo deal Ms. Powers speaks of here:
I have had some friends interviewed about this here:
The top right hand corner of your contract should list dealer as "seller/creditor" and that means financing was in place day one. What dealer did not do, allegedly, is "assign" that contract, to their own detriment. Arguably, the 10 clause is not lawful, but, here, they are beyond the 10 days. They also need to send an ECOA adverse action notice if they deny you credit and they never do. That allows for statutory damages.
You asked if they could repossess. The answer is no, not legally, but, yes, they often do. That just makes it a hotter bed they are in if you end up in a lawsuit.
They made you sign a bunch of papers to protect them, when they sold you the car. Make sure any future dealings are also documented, to protect you. In other words, do not talk to them again, period. Only deal with them in writing and save all writings and voice mails.
Consumer Protection offices (like mine) often offer a free consultation.