John Boehner, the 10-term Republican congressman from Ohio and current House minority leader, seems like a pleasant enough fellow. He enjoys a good round of golf. His voice is smooth and sonorous. His resplendent ocher tan never fades, even in winter. Friends go so far as to call him “the Dean Martin of politics,” a nickname that suggests (correctly) a penchant for boozy bonhomie. And yet now that the GOP’s glowing midterm prospects have positioned Boehner (pronounced BAY-nur) to become the most powerful Republican in Obama’s America—that is, speaker of the House—no one in Washington appears to be particularly happy about it.

Consider last week’s crossfire. On one side were the Democrats, scrambling to trade in their old, tried-and-true bogeyman (George W. Bush) for a more current model (Boehner). First, President Obama traveled to Cleveland, Boehner’s backyard, and knocked the House GOP leader 10 times in a 45-minute speech. Then, on Sept. 11, The New York Times published a story on Boehner’s “tight ties with a circle of lobbyists” that quoted disapproving Dems. By week’s end, the Democratic National Committee had launched two national TV ads and a new Web site slamming Boehner as a cartoonish fat cat—all while blasting out 40 press e-mails to make sure reporters got the point.

On the other side were the Republicans—or at least Republicans of the tea-tinged variety who won last Tuesday’s marquee primaries in Delaware and New York, and will continue to pull Boehner and the rest of the Republican establishment to the right. When Boehner told CBS News that Sunday that he’d be willing to compromise with Obama on the Bush tax cuts—the GOP wants to extend them all; Obama wants to limit them to the sub-$250,000 crowd—conservatives were apoplectic. “This stumble,” wrote The Wall Street Journal, “makes us wonder if he’s ready for prime time.” A new poll showed that 67 percent of Republicans were either undecided on Speaker Boehner or would prefer someone else.

Other grumbling was more discreet. It’s an open secret that Boehner’s Republican lieutenants—Eric Cantor, Kevin McCarthy, and Paul Ryan, who spent last week promoting a new, coauthored book (Young Guns) about how they’re more in tune with the times than their slick establishment predecessors—harbor leadership ambitions of their own. So when they instantly distanced themselves from the boss’s remarks, no one was surprised. As one GOP staffer puts it, “Those guys are pushing themselves forward, with the implication that they’re leaving the rest of the leadership behind. That includes John Boehner.”

The funny thing about all the anti-Boehner ferment, however, is that the Ohioan’s critics may soon come to consider him irreplaceable. In truth, Boehner is one of the few players in American politics with the potential to give both Republicans and Democrats what they need in the wake of November’s anticipated GOP landslide. For the left, that means an experienced legislative negotiator on the opposite side of the aisle. For the right, it means a leader who can rack up tangible accomplishments for the party to run on in 2012—while also keeping the new, red-meat caucus from eating him alive.

It remains to be seen whether the House’s resident Rat Packer will even attempt such a tricky dance step; for now, his tight-lipped staff is content to let everyone imagine the Speaker Boehner they want to imagine. Asked if Boehner would keep battling the White House after November or if, conversely, he’d be willing to negotiate, his spokesman, Michael Steel, left all options on the table—“yep” and “of course” were his wry answers—and refused to elaborate. Whatever Boehner’s intentions, though, his record suggests that he has it in him to become Washington’s “necessary man” in the years ahead—even if both parties consider him little more than a necessary evil.

Every profile of Boehner begins in the same place: the outskirts of Cincinnati, where young John learned the art of compromise (and competition) growing up with 11 brothers and sisters. It’s a nice origin tale. But more revealing is the transformation Boehner underwent between 1991, when he arrived on the Hill, and 2001, when he helped pass No Child Left Behind. The early Boehner was a small-government bomb thrower, not unlike today’s Tea Partiers. He refused to request earmarks, railed against congressional perks, led a drive to abolish the Department of Education, and helped Newt Gingrich craft the “Contract With America.”

But in 2001 Bush came to Boehner, then the head of the House Committee on Education and the Workforce, with a big, expensive plan to significantly increase the federal role in local schools, and the congressman not only went along with it—he ensured its passage. Despite his right-wing résumé, Boehner has said he understood from the outset that “the only way to get a bill was to be bipartisan.” When four-way talks stalled, he shuttled back and forth between the Democratic negotiators, Sen. Ted Kennedy and Rep. George Miller, and his conservative colleague, Sen. Judd Gregg, until they agreed on a compromise. When the White House moved to cut Miller out of the conversations, Boehner threatened to walk. And when GOP leaders, including Dick Cheney, suggested shifting power to the states in an effort to attract more conservative votes, he warned that it was “a deal killer for Democrats.” Boehner’s goal, as The Washington Post’s David Broder wrote at the time, was “to show he was not going to be the partisan he had been in the past.” The plan worked, and NCLB passed.

Boehner’s principles didn’t change at the turn of the century; he’s always been a Chamber of Commerce conservative. But a new administration meant new expectations, and a different set of incentives. Under Clinton, Boehner’s job was to oppose the White House; under Bush, his job was to get stuff done. As a man who made his money selling plastics, Boehner seems to enjoy the latter role more—which is why he broke with many of his fellow fiscal conservatives and backed Dubya on NCLB, Medicare Part D, and TARP, and, more recently, acknowledged reality on the Bush tax cuts. “John is, more than anything, a dealmaker,” says Republican consultant Kevin Madden, a former Boehner spokesman. “He uses his personal and political power to make deals.” (Boehner declined to speak to NEWSWEEK for this article.)

This businesslike pragmatism—he rarely wastes his breath on intractable social issues, for example—has long disturbed conservative absolutists. When Boehner was running for minority leader in 2006, The American Spectator’s Philip Klein told Republicans that electing him would send “a clear signal that they have learned nothing from their electoral defeat and will remain the party of big government.” Boehner’s dealmaking is also a major reason his relationship with the more rigid Young Guns “really is [one] of necessity,” as the GOP staffer puts it.

But it’s also why Democrats might be better off with Boehner leading the opposition than, say, the hyperambitious Cantor. Yes, Bush was a Republican and Obama is a Democrat—meaning that Boehner will be far less acquiescent than he was the last time Republicans ruled the House. But GOP speakers have worked with Democratic presidents before and gotten results. In the mid-1990s, for example, Bill Clinton and Newt Gingrich buckled down and, over the course of dozens of long, wonky White House bull sessions, hammered out plans to reform welfare, balance the budget, pay off $405 billion in debt, provide health care to uninsured children, and make Medicare and Social Security more sustainable—even though Gingrich was better known for partisan warfare than compromise.

Boehner and Obama won’t have the same kind of chemistry, but they will face similar political pressures. If the GOP takes over, the president will have just suffered a major rebuke at the polls, and he will need to prove to a skeptical country that he’s not out of touch. Republicans, meanwhile, will no longer be able to respond to Obama by shouting “hell, no,” as Boehner once did; mainstream voters will expect them to contribute, and will hold them accountable if they don’t. According to Gingrich, Boehner is even better suited to this process—the slow, methodical give-and-take of divided government—than he was. “If I was a roll-out quarterback who occasionally threw an interception,” Gingrich says, “then John is a natural head coach. He knows how to keep the team together and win the game by methodically picking up four yards on every play.”

Of course, if Republicans do capture control of the House in November—current projections show them regaining the majority with 15 seats to spare—many conservatives will insist that the GOP stick to its winning strategy: pure, unadulterated obstruction. It’s hard to imagine 2011’s roiling ranks of Tea Party freshmen, for example, countenancing any kind of compromise with a president they seem to see as a crypto–Muslim socialist, and Boehner may very well assume that he’s better off riding the rage than harnessing it for productive purposes. “Don’t be fooled into believing he’s not a partisan,” warns Miller. “I’ve seen people ask him for a bipartisan meeting, and he says, ‘Don’t waste my time.’” To rule out any future negotiations, however, would be a mistake. Obama has already signaled that his priorities for 2011—deficit reduction, Social Security reform, and job-related tax breaks for businesses—will be GOP-friendly. If newly empowered Republicans refuse to chip in, voters won’t have any reason to reward them in 2012. As Gingrich puts it, “The American people are never satisfied by politicians who fight each other. You can’t be the party of no, because once you’re the majority, ‘no’ just means you’re stopping yourself.”

At a recent lunch with reporters, Boehner complained about dysfunction in the House. “This is supposed to be the greatest deliberative body in the history of the world,” he said. “But there’s very little deliberation [and] very few legislators among the 435 members.” Starting next January, Boehner may finally have the power to change that. The question now is whether he’ll use it.