Schumer, McConnell share a stage in Louisville

US Senate Minority Leader Chuck Schumer, a New York Democrat, at left, presents Senate Majority Leader Mitch McConnell, a Kentucky Republican, at right, with a bottle of bourbon at the University of Louisville's McConnell Center. (Bill Pugliano/Getty Images)

Louisville, Kentucky (CNN)The two top leaders of the Senate sought to paint a picture of civility in a rare dual appearance outside the Capitol, following a year of bitter fighting between the two parties in Congress over issues like tax cuts, health care and the Supreme Court.

Senate Majority Leader Mitch McConnell introduced his Democratic counterpart, Minority Leader Chuck Schumer, before a speech Monday morning that Schumer gave at the McConnell Center at the University of Louisville.
At the top of his speech, Schumer presented McConnell with a bottle of bourbon that was made in Brooklyn, where Schumer was born and raised. "I'm sure he'll never agree that New York bourbon even counts as bourbon," Schumer quipped.
Toward the end, the McConnell Center presented Schumer with a clock that has the university's emblem on it. "Story of my life: Mitch gets the bourbon, I get the clock," Schumer joked.
    It's not unusual to see the two men together. They speak on a near daily basis on the Senate floor when they each deliver their opening remarks, and they are frequently involved in negotiations over legislative action that takes place on the floor.
    But it's not a common sight to see them "take this show on the road," as McConnell teased at one point. While the two men acknowledged their differences, they attempted to keep the message focused on the recent two-year budget deal they struck and the upcoming immigration debate this week on the Senate floor.
    The Senate will vote Monday night to open up debate on a bill that will serve as the main vehicle for immigration legislation. Members can propose amendments to the bill, but they must get at least 60 votes to pass -- a high hurdle that will require some bipartisanship, as the chamber is currently split 51 Republicans to 49 senators who caucus with the Democrats.
    Schumer gave McConnell credit for holding an open floor debate on immigration, and he framed the Senate as the best chance to carry out "the business of the country," describing the House as too "fractious" and saying with a shrug "the President is the President."
    He urged Trump to "lead in a bipartisan way" and recalled his own involvement in the "Gang of Eight" immigration bill, saying many of the policies that people currently want were in that bill. 
    McConnell (right) (R-KY) and Schumer (D-NY) shake hands after Shumer delivered a speech and answered questions at the University of Louisville's McConnell Center on Monday in Louisville, Kentucky.  (Bill Pugliano/Getty Images)
    Schumer, however, didn't hold back in his criticism of the other senator from Kentucky, Rand Paul. He was asked during a Q&A portion by a member of the audience about Paul's opposition to the massive spending bill, which is expected to add to the deficit.
    He said Paul's concerns about fiscal responsibility "ring very hollow" because he supported the tax bill that also increases the deficit. "If you're a deficit hawk, you got to be a deficit hawk all the way through," Schumer said.
    Asked about Democratic strategy for the 2018 midterms, Schumer said his party can't rely solely on public resistance to the President.
    "Our mistake, I think, you cannot just run against Donald Trump. It is the job of we Democrats to put together a strong cohesive economic group of proposals aimed at the middle class," he said, pointing to a rollout of policy proposals Democrats have been making since last summer.
    This was the second time the McConnell Center has had both Senate leaders on the stage. The last time was in 2007 when former Sen. Harry Reid was a McConnell Center guest.
    When McConnell introduced Schumer, he said the center has had many interesting guests over the years but "none more interesting than our guest this morning."
    The two men argued that their relationship was closer than it appears. "We're kind of like the offensive and defensive coordinators," McConnell said, joking that offensive coordinator is a better position.  
    "Robust debate is not unusual," McConnell continued. "But at every critical moment in this country, we've come together to do what is needed to be done to move the ball down the field."
    "We really do get along despite what you read in the press," Schumer said.
    There was no mention of Schumer voting against McConnell's wife, Elaine Chao, in her nomination to become transportation secretary early last year, which was seen in the media as a major sign of partisanship in the immediate aftermath of Trump's election. Nor was there in-depth discussion of the all-consuming legislative battles that rocked Capitol Hill in recent years.
    Weighing in on what he sees as shared value with McConnell, Schumer said both men consider it their jobs to unify their respective members.
    "We have very diverse parties, he has a Susan Collins and Ted Cruz. We won't say Rand Paul, we're in Kentucky," Schumer said to laughs. "I have Bernie Sanders and Joe Manchin. I think what makes us most successful ... is to try and create unity in our caucuses."