Former Vice President Dick Cheney speaks at a cardiovascular symposium in McLean, Va., on April 27, 2012. Cheney will host a fundraiser at his home for Mitt Romney on Thursday. (Manuel Balce Ceneta/Associated Press)

When Mitt Romney arrives Thursday at the gates of Teton Pines, a majestic Wyoming country club where captains of industry flock each summer to golf on an Arnold Palmer-designed course, his purpose will be greater than spending another evening separating rich people from their money.

The presidential candidate will be taking a big step toward becoming the official head of the Republican Party, as he is feted at the club and then at a $30,000-a-couple dinner at the nearby home of Richard B. Cheney, the living thread connecting the past five GOP presidencies.

By hosting the fundraiser, the former vice president — who in his retirement remains a powerful leader of foreign policy neoconservatives yet a deeply polarizing figure outside of the Republican base — will make his grandest gesture to pass a torch to Romney.

Prominent conservative rainmakers are joining Cheney in holding the event, including Foster Friess, an investment manager who bankrolled a pro-Rick Santorum super PAC that attacked Romney during the GOP primaries; John R. Miller, a beef company executive; and L.E. Simmons, president of a private-equity firm with large oil investments.

It was unclear how many of Cheney’s political allies would appear at the dinner, but the hosts also include Dick Scarlett, a banker and one of Cheney’s closest friends and fishing buddies, and Cheney’s eldest daughter, Elizabeth, a former State Department official.

Thursday’s dinner at Cheney’s home in Wilson, just outside Jackson Hole, coupled with an earlier reception at the Teton Pines is expected to raise more than $2 million, according to a top Romney fundraiser.

The evening will put a spotlight on Romney’s relationship with the previous Republican administration, which has been complicated and not always comfortable.

Romney advisers characterized his relationship with Cheney as cordial, but no deeper than one that any elder statesman would be expected to have with his party’s presumed presidential nominee. They speak infrequently, and advisers said there is little evidence of Cheney’s influence, or that of Cheney’s close associates, on Romney’s policies or politics.

Romney also has a distant relationship with former president George W. Bush. The two speak by phone occasionally, but some advisers who bridge the Romney and Bush-Cheney campaigns said Bush does not want to intrude on Romney.

“This does not look to me like Bush-Cheney redux,” said former congressman Vin Weber (Minn.), a veteran of the Bush-Cheney campaigns and a senior policy adviser to Romney.

“At the broader advisory level, everybody who was around Cheney and Bush are around Romney,” he added. “They want him to win. And it’s inevitable that they’d have some influence, because they have the most recent Republican expertise in running the government. But I don’t see a lot of overlap there.”

Romney’s relationship with former president George H.W. Bush, however, is more extensive and personal. Romney delivered a major speech on Mormonism at Bush’s presidential library four years ago and visited the former president this spring in Houston, while former first lady Barbara Bush recorded robo-calls for Romney during the primaries.

In this campaign, though, Romney has not appeared in public with George W. Bush or Cheney, making his visit to Wyoming noteworthy.

People who know both Romney and Cheney said they have contrasting leadership styles. Where Cheney comes off at times as sharp-tongued, Romney often projects a sunny optimism and sometimes seems uncomfortable on the attack. Where Cheney’s beliefs and policies are rooted in conservative ideology, Romney’s tend to be driven by analytical problem-solving.

Yet these people said they see similarities in their worldviews as well as other mannerisms. Both are hawkish on foreign policy, for instance, voicing unrepentant support for U.S. efforts to combat terrorism and beef up the military.

“They both very much believe in a strong defense and exerting American power where need be,” said Tom Sansonetti, a close friend of Cheney’s who served in the younger Bush’s administration and is a Romney donor.

“The way that [Romney] has addressed some of the international issues that have occurred, such as Libya, his attitude and response toward Iran and their attempt to attain nuclear weapons, his approach to Syria — if it were candidate Cheney instead of candidate Romney, I think they would have the same view on those matters,” Sansonetti added.

One George W. Bush administration official who has worked closely with Cheney and Romney said they are “very different.”

“They’re both very probing personalities in the context of policy discussions. They don’t let up. They really drill down. They’re both very smart and curious intellectually,” said the official, who spoke on the condition of anonymity to be frank.

“They’re different in that Cheney really keeps his cards close to his chest, even in policy meetings, whereas Romney is not as focused on keeping you guessing,” the official added. “Romney has a style of more freewheeling, open discussion. He isn’t worried about asking dumb questions.”

Many Cheney allies who shaped policy in the Bush years — including Lewis I. “Scooter” Libby, Paul Wolfowitz, John C. Yoo and David Addington — have no roles in the Romney campaign. Nor do many senior foreign policy figures from that period, such as Condoleezza Rice, Donald H. Rumsfeld, Colin L. Powell, Robert M. Gates and Stephen J. Hadley — although Hadley endorsed Romney in April and Rice spoke at Romney’s donor retreat last month.

“Romney’s his own man and brings his own approach,” said Charlie Black, an adviser to the candidate.

Still, many political operatives or lower-ranking policy officials from the George W. Bush administration are deeply enmeshed in the Romney campaign, including senior adviser Ed Gillespie; foreign policy adviser Dan Senor; economic policy advisers Glenn Hubbard and Gregory Mankiw; and strategists Stuart Stevens and Russ Schriefer. And several senior Romney staff members are veterans of Bush’s Republican National Committee, including campaign manager Matt Rhoades.

Asked to assess their relationship, Tom C. Korologos, an ambassador to Belgium in the Bush administration, echoed the thoughts of several other Republicans who know Cheney and Romney.

“I’m not sure they have any relationship except that they are of the same party,” Korologos said. “I never see any signs of Cheney people around the Romney campaign.”