WASHINGTON — The relationship began in the most Washington way ever: on the set of C-Span.

Will Fischer, then the director of government relations for VoteVets, the liberal political action committee, was tapped to face off with Dan Caldwell, the executive director of the conservative Concerned Veterans for America. It was a continuation of a yearslong and contentious dialogue over veterans issues, including disputes over health care, which candidates care more about matters important to veterans, as well as their dueling views on the nefarious nature of the Republican or Democratic parties.

But then the two found an unanticipated policy bridge, and have now gone on to work together to persuade Congress to finally revoke authorizations of military force passed after Sept. 11, 2001, which both believe have been bent and stretched to justify wars far beyond Congress’s intentions nearly two decades ago.

“I honestly did go into the interview expecting a combative conversation,” Mr. Caldwell said, “but when we started talking about foreign policy, it was clear there were some areas of alignment especially on war powers. The wheels started turning in my head, and we came together and decided to pursue some of these shared goals.”

Both groups — who are wolf to the other’s sheepdog on nearly every other policy issue — intend to share a legislative agenda this year that presses for changes to war authorization measures and an end to the United States’ presence in Afghanistan. They plan to more strongly tie their substantial financial and news media support in 2020 to candidates’ views on foreign policy.