A year ago, Senate Minority Leader Charles E. Schumer voted against the confirmation of Transportation Secretary Elaine Chao, whose husband just happens to be Majority Leader Mitch McConnell.

A decade ago, Schumer (N.Y.) recruited and counseled McConnell’s opponent in what  became the closest reelection battle the Kentucky Republican has ever faced.

Coming off their work to pass last week’s huge bipartisan budget deal, Schumer and McConnell appear to be trying to put those contentious moments behind them. This week, they are overseeing a wide-open debate on immigration and border security, a throwback to when the Senate used to be a more deliberative, and bipartisan, chamber.

This all led to Monday’s perfectly timed distinguished lecture series at the McConnell Center at the University of Louisville, the majority leader’s alma mater, where he gave a warm introduction to his rival.

“Actually, the Senate is a pretty collegial place. We don’t dislike each other,” McConnell said. “We have to work together.”

Schumer brought his host a bottle of Widow Jane, a bourbon that is distilled not far from his home in Brooklyn. For Kentucky purists, bourbon is not bourbon if it is not produced in the Bluegrass State. It is whiskey. McConnell posed for pictures with Schumer’s peace offering and then smiled as the New Yorker called the budget deal a building block for future compromises.

“It showed that, even in very divisive political climate, the Senate can be the place where the business of the nation gets done,” Schumer said.

Democrats have not often felt that way about McConnell. He refused to consider Judge Merrick Garland’s 2016 nomination to the Supreme Court and spent most of 2017 pursuing entirely partisan initiatives — the failed effort to repeal the Affordable Care Act and the successful, party-line vote to cut taxes by $1.5 trillion.

Historically, the relationship between Senate leaders is one of the most important in Washington. The chamber’s unique rules produce bipartisan compromise that often forces action in the more partisan House.

A great bond between the two Senate leaders can produce a bounty of important legislation, with the best example coming in the 1960s as Democrat Mike Mansfield (Mont.) and Republican Everett Dirksen (Ill.) oversaw passage of sweeping civil rights laws. If the two leaders are in constants squabbles — as was often the case in the past few years of McConnell’s service next to Harry M. Reid (D-Nev.) — it can turn the Senate into a legislative graveyard.

The McConnell-Schumer relationship will surely be tested in the months ahead, and a true judgment might not come for years, but this week’s immigration debate will be another big moment. Schumer’s liberal base is clamoring for legislation giving a path to citizenship to “dreamers,” the hundreds of thousands of undocumented immigrants brought here as children.

McConnell has been fairly circumspect about his own views, but he faces a right flank demanding tough border security and restrictions to legal immigration.

“We have proven the Senate can function when both parties work in a bipartisan way,” Schumer said Monday, referring to the budget deal. “This week the Senate will have an opportunity to build on that progress.”

Schumer, 67, is just 13 months into the role of Democratic leader while McConnell, 75, is in his 11th year, set to become the longest serving Senate GOP leader in June. They are two very different personalities. Schumer is voluble and publicly emotional, while McConnell is reserved and stone-faced.

Advisers to both leaders note they have one key attribute in common: a political instinct that comes with having both served as campaign committee chairman in their respective parties.

Each knows his first job is to protect the senators in his caucus. That is why McConnell was so confident during last month’s brief government shutdown that Schumer would cave, because the issue was playing poorly in five conservative states where Democrats are up for reelection this year. Any chance of winning the majority in November begins with protecting those red-state Democrats.

It is exactly how McConnell behaved in 2016 when he was defending a collection of incumbents up for reelection in Democratic-leaning states.

In other words, McConnell and Schumer understand each other’s core motives.

Democrats say Schumer has gone out of his way to be open and direct with McConnell, in part because the Reid-McConnell relationship fell apart amid a lack of trust and communication. Reid would say only part of what he was thinking, and McConnell could be unresponsive.

On Monday, Schumer reiterated his proclamation that, left to their own devices, without President Trump’s interference, he and McConnell could reach a deal even on immigration. He cited a couple things he and McConnell agreed to early on: “Never ask things that are impossible of the other, to be honest and respectful.”

Schumer’s voted against Chao, who had served as labor secretary in the George W. Bush administration, came early last year, a few days after Trump announced his first travel ban from countries with large Muslim populations, a move that prompted Schumer to declare his opposition to many Cabinet choices.

Schumer spoke to McConnell privately about the matter and viewed it as water under the bridge, but it lingered in McConnell’s mind, according to advisers for both men. He wondered whether Schumer was too afraid of the anti-Trump base and was making a political move, reverting back to his days as chairman of the Democratic Senatorial Campaign Committee.

Schumer was brutally effective in a four-year run, bringing Democrats to the cusp of a filibuster-proof, 60-seat majority after the 2008 campaign. One seat he did not win, McConnell’s in Kentucky, left the Republican with a bitter feeling toward Schumer.

They had been part of a bipartisan group that crafted the Wall Street bailout that year, but in the final weeks of the campaign McConnell faced ads critical of the deal.

Now, Schumer wants McConnell’s job, and he wants to be majority leader, and those coming elections may serve as the biggest test of their relationship. Schumer expected Democrats to win the majority in 2016, when he replaced Reid. After university officials gave him a Louisville clock as a parting gift, he joked that McConnell again got the better end of the deal.

“Story of my life,” he said. “Mitch gets the bourbon. I get the clock.”

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