One of the ironies of the Republican takeover of the Senate is that it has made President Barack Obama more of a central player in Washington’s legislative debates than he had been before. In the last Congress, when the House was under Republican control and Democrats ran the Senate, the House Republicans easily passed dozens of bills on party-line votes that were then ignored by the Senate. When the Senate occasionally mustered the ability to advance legislation, the House repaid the favor. Obama was often a bystander as the two chambers produced gridlock.

Obama learned that publicly campaigning for the few bills that did have some life, like comprehensive immigration reform, actually made them less likely to pass. At one point during the immigration debate, Democratic senators begged Obama not to make a high-profile speech on the issue. And so, increasingly, he remained publicly absent from the negotiations. Nobody was really in charge and nothing got done: it was the least productive Congress in modern history.

The lines of authority are a little clearer now, even if John Boehner, the Speaker of the House, and Mitch McConnell, the Senate Majority Leader, have the continuing challenge of dealing with their right flanks. They also have a strong incentive to work together to move legislation: going into 2016, they want to show that the G.O.P. is unified and capable of governing. But after those bills pass both chambers, they will reach Obama’s desk. Obama and his veto have replaced Harry Reid, who has now been reduced to the role of minority leader, as the crucial blocking tool for Democrats.

The first test of whether this new dynamic will lead to a more productive Congress comes this week, with the debate over Keystone XL, the Canadian pipeline project that has been mired in political controversy for six years. Republicans in both the House and Senate declared that approving Keystone XL, which would carry oil from Alberta, Canada, to U.S. refineries in the Gulf of Mexico, would be their first order of business, and the House approved its version of the legislation on Friday afternoon.

At first glance, it’s absurd that approval of a Canadian corporation’s pipeline project is the new Republican Congress’s highest priority. The benefit to the U.S. economy from the construction of the pipeline would be modest—a few thousand temporary jobs created during its construction and a few dozen permanent jobs once it’s up and running. Republicans have long argued that the pipeline would help reduce fuel prices, a tenuous claim that is now beside the point because the price of a gallon of gas is at a five-year low.

Still, there is a scenario in which Boehner and McConnell’s decision to start with Keystone might be encouraging. The Obama Administration immediately announced that the President would veto the legislation. But the veto statement was silent on the merits of the project itself. Obama simply objected to Congress interfering with the pipeline-approval process, which has historically been left to the executive branch. The President’s exclusive right to approve cross-border pipelines may be such a fundamental principle to the Administration that the veto statement is the beginning and end of the debate. (Republicans don’t have the necessary sixty-seven votes to overturn an Obama veto.)

But what if the White House saw the fight over Keystone as an opportunity for a larger deal? Keystone XL is one of the few G.O.P. priorities in which the philosophical gulf between Obama and congressional Republicans is relatively narrow. In private, Obama has been dismissive of environmentalist claims that building Keystone XL would significantly affect climate change, and his State Department, with some caveats, came to the same conclusion in an environmental-impact statement. In U.S. and Canadian diplomatic circles, officials regularly discuss whether there could be some kind of a deal between the two countries. For example, Canada might make a more ambitious pledge to reduce carbon pollution in return for U.S. approval of the Keystone XL pipeline.

So why doesn’t Obama have that discussion with Congress instead? What would the G.O.P. be willing to trade to get Keystone approved? A carbon tax? A large infrastructure project? Codifying the E.P.A.’s climate regulations into law? From the White House’s perspective, the Keystone XL pipeline should be an ideal policy to give away in a trade: it’s a major issue that Republicans care a great deal about but one that Obama seems to view as a sideshow. (And if world oil prices remain low, Keystone may be entirely moot because production in the expensive-to-develop Canadian oil sands might not be economically viable.)

Obama’s final decision on the matter is fast approaching. The last major bureaucratic step is a State Department review of the “national interest” in the project, which had been postponed while a Nebraska court considered a challenge to the route of the pipeline through that state. On Friday, the court ratified that route, clearing the way for the State Department to complete its work. It is possible, and perhaps even more likely than not, that soon after Obama vetoed congressional legislation forcing approval of Keystone XL he would then approve the project on his own. But will he get anything in return? If the Republican Congress and President Obama can’t figure out a way to cut a deal on this issue, don’t expect the next two years to look much different from the last two.