In Kansas recently, Republican Senator Pat Roberts, who’s in a tough race for reelection, made a statement that left me puzzled. “A vote for me is a vote to change the Senate back to a Republican majority, and we’ll get things done,” he said. “And it means a stop to the Obama agenda.”

Wait a minute, I thought. Which is it—ending the status quo of Washington gridlock? Or ratcheting up the gridlock by obstructing President Obama? You can't "get things done" in Washington without the president's signature, and no matter what happens in this year's elections, he's not going anywhere for another two years.

Yet these two seemingly contradictory messages are at the heart of Republican Senate campaigns across the country. I’ve heard them from candidate after candidate. And the paradox behind them gets to the question political watchers are increasingly pondering: If, as seems likely, Republicans take the Senate, what then? Will the GOP see its takeover as a mandate for ever more extreme partisanship? Or will the party suddenly turn conciliatory, ushering in a new age of progress? A new Republican Senate majority will put the party at a crossroads as it tries to reconcile these two competing promises.

One possibility is that nothing will really change. After all, we have divided government now, and we will still have divided government if Republicans go from 45 senators to 51. Obama will still be in the White House, and the House of Representatives will still belong to the GOP. With the Senate requiring a 60-vote supermajority for most legislation, Republicans have effectively had a veto in the upper house since Scott Brown was elected in 2010. Democratic priorities like gun control or a minimum-wage hike wouldn't be any deader in a Republican-controlled Congress than they already are. The status quo is often a good bet in Washington, and it may well be that it simply continues.

A Republican Senate majority would still change a few things, however. It would make it harder for Obama to get his nominees approved, something Democrats have been able to do with just 51 votes since they changed the Senate rules with the “nuclear option” last year. It would make Mitch McConnell the majority leader, with the power to decide what legislation gets considered and how it proceeds through the legislative process. Given McConnell’s track record of keeping Republicans staunchly unified against virtually anything Obama proposes, many observers, particularly liberal pundits, believe McConnell would quickly devote the Senate to passing a raft of partisan legislation that Obama would never allow to become law: repealing Obamacare, defunding Planned Parenthood, approving the entitlement-slashing House budget plan authored by Representative Paul Ryan, restricting the Environmental Protection Agency, and so on. The result would be a more partisan, toxic, and stalemated Washington than ever before.

There’s evidence to support this view. In an interview with Politico in August, McConnell said he planned to “challenge” Obama by passing spending bills that included “a lot of restrictions on the activities of the bureaucracy.” That is, Republicans would attach their policy priorities—McConnell specifically mentioned reining in the EPA—to the legislation that funds the government, forcing Obama either to approve their pet projects or shut down the government. On the trail, GOP candidates like Roberts frequently make reference to the bills the House has passed that the Senate hasn’t taken up. There are 387 of them—you can find a full list here—ranging from the “Permanent Electronic Duck Stamp Act of 2013” to bills that would expand offshore oil drilling and get rid of all or part of the Affordable Care Act. Many are what’s known as “messaging bills”—legislation that House Republicans knew would never become law and passed just to make a statement about their priorities, or to satisfy a constituency. Sending those bills through the Senate would result in a speedy Obama veto, accomplishing nothing.

But those who see McConnell only as an obstructionist are overlooking another significant part of his profile: his record as a dealmaker. As the general election nears, McConnell has sought to emphasize this as well. “There have been three major bipartisan agreements during the Obama years between Republicans and Democrats,” he said in last week’s Kentucky Senate debate with his Democratic opponent, Alison Lundergan Grimes. “The vice president and I have negotiated every one of them.” McConnell was referring to the December 2010 deal to extend the Bush tax cuts, the last-minute 2011 deal to raise the debt ceiling, and last year's fiscal-cliff deal. This is the side of McConnell that drives conservatives crazy, putting him in the unique position of being ardently reviled by left and right alike. But as Alec MacGillis’s excellent new ebook on McConnell makes clear, the Kentucky senator’s top priority has always been not ideology but his own political advancement and survival. He made those deals because, much as his base hated to see him working with Democrats, the alternative would have been even worse for the GOP—and him—politically.

When and if they take control of the Senate, Republicans will have a big incentive not to simply create more gridlock: It would make them look terrible, worsening their image as the “party of no” and making it harder for their presidential nominee to win in 2016. The same goes for passing unpopular legislation like the Ryan budget or repealing Obamacare—which most voters do not favor, even though the law is also unpopular. As things stand today, neither party is to blame when Congress can’t get anything done, because each party controls half of the Capitol. (While the House’s dysfunction is well known, the Senate has also become a legislative graveyard, to the point that even Democratic senators publicly complain about it.) But with control of both houses of Congress, Republicans would be on the hook for Congress’s actions. They alone would get the blame if Congress remained dysfunctional—and they alone could claim credit if Congress actually passed bills with popular support. If Republicans passed such moderate, constructive legislation, Obama would be hard pressed to simply veto everything they put on his desk.

“The way I describe it is, we’re putting the guardrails on the Obama administration’s last two years,” Senator Rob Portman told me in a recent interview, explaining how he envisions a Republican-controlled Senate proceeding. Needing GOP approval for nominees, Obama would have to appoint moderates to judicial and executive positions, he said. But Portman, a fiscally focused Ohio Republican who is generally conservative but believes in bipartisan compromise, sees several areas of potential cooperation with the administration. He mentioned tax reform, a “grand bargain” on the budget, an energy bill—perhaps something that combines Keystone XL pipeline approval with reductions in carbon emissions—and new free-trade agreements, which Obama has supported but Senate Majority Leader Harry Reid has blocked. Portman, who voted against the bipartisan immigration-reform bill that passed the Senate last year, also believes a Republican-led immigration-reform bill could pass the House and Senate and potentially be approved by Obama.

Portman rejected the idea that Republicans are contradicting themselves when they promise voters that they’ll stop Obama while also ending gridlock in Washington. “There’s not an inconsistency between saying there’ll be more oversight and more limitations on what the president could accomplish, and at the same time we’ll move forward positive legislation to get the economy back on track,” he said.

Essentially, Republicans will stop the Obama agenda by proposing their own agenda—one that’s realistic and palatable enough that Obama will agree to it. Portman has been pushing his party from within to offer such an agenda for the sake of the next GOP presidential candidate. (Portman himself has been mentioned as an aspirant, a possibility he told me he’s considering but not discussing until the midterms are over.) “We need the majority for the sake of the country for the next two years, but it’s also about having a proactive, positive Republican agenda to take to the American people in 2016,” he said.

This possibility—that a GOP eager to prove it can be constructive and govern will seek compromise with Obama—could make a Republican-controlled Senate the best thing to happen to the lame-duck president, as The Wall Street Journal’s Gerald Seib has written. But it is easier said than done. Even if Republican leaders want this to happen, the biggest obstacle will be Republicans themselves—chiefly the restive conservatives in the House, who have prevented consideration of bipartisan legislation approved by the Senate on issues like immigration and who have often prevented the party from doing what's in its own political interest (see: government shutdown). McConnell's Senate majority will include pragmatists like Portman—but also ideologues like Ted Cruz. Portman acknowledged this obstacle when I spoke to him. “We as Republicans have a real challenge to get the diversity of our ranks to work together,” he said.

Individual issues are also sure to prove harder in practice than in theory. Tax reform, which would involve taking tax breaks away from the interest groups and armies of lobbyists that ardently support them, gets lip service from both parties, but when Republican Representative Dave Camp unveiled an actual tax-reform plan this year, everyone, including Republicans, immediately abandoned him. (Camp, who had worked on the proposal for years and hoped it would be the capstone of his 24-year legislative career, was so discouraged by its reception he announced his retirement shortly thereafter.) The same goes for entitlement reform, which requires both parties to agree to a combination of politically toxic measures—tax hikes and cuts to Social Security and Medicare—with virtually no constituency outside the centrist pundits who cheer for such things. (Sure, it would reduce the deficit, but that’s already happening anyway.) In forums like the 2011 supercommittee, Democrats have offered to agree to major cuts, but Republicans have been unwilling to compromise in turn by agreeing to increase revenue.

At least in the abstract, however, there are a number of bills a Republican majority could pass that Obama would agree to sign. Obama—the real Obama, not the left-wing warrior of conservative fever dreams—loves the idea of bipartisanship and has been frustrated by a GOP he sees as unwilling to come to the table. He has agreed in principle, in the past, to ideas like the grand bargain, which his base loathes. Liberals also suspect Obama is willing to allow the Keystone pipeline, a decision on which he has delayed in the face of intense pressure from environmentalists. Most liberals contemplating a GOP Senate majority have focused their preemptive ire on the image of a vengeful McConnell threatening more brinksmanship and shutdowns. But perhaps it’s the dealmaking McConnell they should fear more.

Some, in fact, are already worried about this. I recently asked a top Democratic strategist why he worried about a Republican Senate takeover when, after all, McConnell would still need Democratic votes to pass legislation and Obama could still block bills with a veto. “What scares me the most,” he said, “is what Obama will agree to."