Moderated by Ryan C. Crocker, the United States ambassador to Baghdad during the 2007 surge of American forces in Iraq and now the dean of the Bush School of Government here, the men opened by so praising one another’s “splendid” and “masterful” past actions that Mr. Crocker dryly remarked, “What I’m trying to do is find out if you’re all eligible for sainthood.”

Mr. Crocker then asked, “Were there ever serious disputes among you?”

The question prompted Mr. Cheney, whose voice seemed weaker after heart surgery last summer, to speak up for the first time. “I remember a big argument over the question of whether we needed to go to the Congress to get authorization for this war,” he said. His position was, he recalled, “Look, if we ask for approval and they don’t give it to us, then what are we going to do?”

History now knows he lost the argument  the president wanted approval from Congress and got it  although the men on stage agreed that Mr. Bush would have gone to war without authorization from Capitol Hill. (Mr. Bush had at this point moved to the front row of the audience.)

Mr. Cheney’s recollection prompted Mr. Powell to recall his own arguments with Mr. Cheney.

“There were occasional moments when he thought I was off the farm,” Mr. Powell said of Mr. Cheney, who as defense secretary was the civilian leader in charge of the military, including Mr. Powell. “I generally knew when I had stepped over the line and he quickly reminded me I had stepped over the line.”

Mr. Powell and Mr. Cheney went on to have far more venomous struggles before the Iraq invasion of 2003  Mr. Cheney pushed for war, Mr. Powell warned against it, then went along  and their appearance side-by-side on stage was one of their rare, if only, public moments together in the years since.