If I traded cars with someone and they want their old car back, do I have a right to keep the car?

As a general matter, sales--especially sales between private parties who are not normally in commerce (and so not subject to some of the rules governing commercial entities or retail establishments) are final, unless and only to the extent that--

* The agreement of sale provides for refunds, returns, etc.

* Or there was fraud in the agreement or sale, which would provide grounds to rescind (void or cancel) the contract.

A swap is a sale--you are exchanging car A for car B, and legally, it is no different than exchanging a car for money. Assuming there was no fraud and no refund, etc. provision in the agreement, you can keep the car. Of course, if the other party is willing to try to take you to court, you might ask yourself whether--even if you're legally in the right and would ultimately win--it's worth the potential cost and headache to fight.