SAN FRANCISCO — The Winklevoss twins went to the Ninth Circuit U.S. Court of Appeals today hoping to undo a settlement that makes them far richer than it did when they struck the deal with Facebook in 2008.

So perhaps it wouldn’t be too big a blow if the appeals panel were to reject their claims — as it appeared inclined to do — and uphold an agreement that gives the twins and their business partner in ConnectU $20 million in cash and 1.25 million shares of private Facebook stock once valued at $45 million. The Winklevosses, Cameron and Tyler, took front-row seats at the latest showdown in their years-long legal battle, which has played out on both coasts and even made its way into a feature film. The heart of their suit — that Facebook founder Mark Zuckerberg stole their idea after hiring him to work on their social networking site — is what they want the chance to pursue anew.