“All right, I’ve had enough!” Tracy Patterson’s old-school Brooklyn accent rang out in a bar in Arizona where a New York Jets fan club had gathered to witness its team losing — again. The fan club meets for every Jets game at the same bar, a masosistic ritual of perennial disappointment. But this season, the heavy burden of being a fan of the Green and White comes with a twist: Two people associated with the team — a retired kicker, Jay Feely, and a former Jets sideline doctor, Amish Shah — won their respective primaries and will face off for a U.S. House seat representing the Scottsdale area.

Mr. Feely is a Republican; Dr. Shah is a Democrat. “The New York Jets are guaranteed at least 1 win in November this year,” wrote one jaded fan on social media. The race is no joke, however: Arizona’s First Congressional District is considered a coin toss that could help determine control of Congress. Although their tenures with the Jets overlapped, the two men don’t remember meeting each other. Still, when Dr. Shah won his primary, Mr. Feely called to praise his civil approach.

That civility had a short shelf life, with Mr. Feely accusing Dr. Shah of aligning with far-left Democrats. Sharp differences include health care — Dr. Shah supports a Medicare option for all ages, while Mr. Feely opposes government involvement — and Israel policy. Both portray themselves as independent-minded in a state famous for producing maverick politicians like Barry Goldwater, John Mc Cain, and Kyrsten Sinema.