He’s still a Republican and has not endorsed Barack Obama for president, but Sen. Chuck Hagel of Nebraska’s presence at Obama’s elbow during last week’s tour of the Middle East — peering out of a helicopter over Baghdad, chatting with soldiers at Camp Eggers in Afghanistan — lent a bipartisan, fact-finding air to what was otherwise an Obama campaign trip.

Just by accompanying Obama and Democratic Sen. Jack Reed of Rhode Island, Hagel has done for Democrats what independent Sen. Joseph I. Liebermanof Connecticut has done for Republicans by crossing into the opposition camp, if only to visit.

Hagel is the second-ranking Republican on the Foreign Relations Committee, and his presence as Obama met with Afghan President Hamid Karzai and Army Gen. David H. Petraeus enhanced what political strategists call the “optics” of the trip, a visual impression of bipartisanship and gravitas.

“Hagel provides a physical, political reminder that Obama has been looking over the horizon at these issues,” said Michael Feldman, a senior adviser to Al Gore when he was vice president and now a partner with the Glover Park Group, a strategic communications and lobbying firm. “Hagel being along on this trip is a reminder that there is very broad-based, bipartisan support for Obama’s approach.”

Hagel has complimented Republican Sen. John McCain of Arizona as well as Obama, and the two Republicans remain friends. But Hagel’s common ground with Obama is based on his opposition to the war in Iraq.

Hagel is a combat veteran of Vietnam who voted in 2002 to authorize military action against Iraq. He later changed his mind, though, and became one of the strongest Republican critics of the Bush administration’s handling of the war. By 2005, he was calling for withdrawal of troops and more debate on the war in the Republican-controlled Congress. “To question your government is not unpatriotic,” he told the Council on Foreign Relations that year. “To not question your government is unpatriotic.”

He and Gordon H. Smith of Oregon were the only GOP senators to defy a White House veto threat in March 2007 and vote for withdrawing most U.S. combat troops from Iraq within a year.

Popular with Democrats as well as Republicans in Nebraska, and, like McCain, something of a maverick, Hagel considered running for president himself but rejected the idea and decided to leave the Senate at the end of this year.

His name came up right after Bush’s re-election in 2004 as a potential Republican successor, and a group of students at Harvard’s Kennedy School of Government even predicted in a 50-page analysis that Hagel would be elected in 2008 — with former Massachusetts Gov. Mitt Romney as his running mate.

That paper showed the perils of political forecasting: The students assumed a “conservative realignment of the nation” and an economic upswing and figured Hagel and Romney would defeat a Democratic ticket of Sen.Evan Bayh of Indiana and former Virginia Gov. Mark Warner. Obama, who had been elected that year, did not figure in; nor did McCain.

This year, Hagel has been mentioned as a possible vice presidential choice for Obama. (And Romney is widely believed to be on the McCain short list, with an announcement seeming likely this week.) Hagel has said he would consider the idea, though he doesn’t think he would be asked. Neither do Feldman and others who have followed the race.

Though Hagel is identified as a moderate, his differences with Bush and other Republicans are mainly on defense and foreign policy issues. During the Bush presidency, from 2001 until the start of last week, he joined his caucus 88 percent of the time when most of them voted against most Democrats; he’s taken Bush’s side 90 percent of the time. Both scores are close to the Senate GOP median. Hagel’s opposition to abortion, though, would probably disqualify him from the Democratic ticket. A Cabinet post would be more likely.

Even if Hagel has no future role with Obama, the two have a rapport now. Obama respects Hagel’s “independence and willingness to think critically about national security issues,” says Rep. Artur Davis, who’s close to Obama.

“In some ways,” the Alabama Democrat says, “Sen. Obama developed a relationship with Sen. Hagel that he expected to develop with Sen. McCain. Of course, presidential ambitions got in the way of that.”