WASHINGTON » He offered to explain the intricacies of the federal budget. She reminded him that she had majored in math.

From the moment they knew they would be colleagues, representing the same state, it was clear the relationship would not be a great one.

Meet the Senate’s oddest couple: Ron Johnson, Republican, and Tammy Baldwin, Democrat, of Wisconsin, the personification of this polarized Congress, right down to their mahogany desks at opposite ends of the Senate chamber. No two senators from the same state vote against each other more often than they do – 75 percent of the time in 2013, a review of Senate records shows.

And when they do find patches of common ground it is often on uncontroversial nominees to lower courts and innocuous legislation like a bill to privatize the federal helium reserve. He, an uncompromising fiscal conservative, had never run for anything before the Tea Party wave of 2010 helped sweep him into office. She, a member of the House for 14 years who championed universal health care and stricter financial regulations, got her start in politics while still in law school.

One of the 10 richest members of the Senate, he started a business that makes plastic packaging for medication and lunchmeat in Oshkosh, the quaint working class-city where he lives. She is of more modest means, makes her home in the liberal bastion and capital city of Madison and is the first openly gay person to serve in the Senate.

He has earned a 100 percent rating from the American Conservative Union and the endorsement of the National Rifle Association. She has "A" grades from the Sierra Club and the National Education Association, the teachers’ union. And while he filed a lawsuit against the Obama administration last week over the Affordable Care Act, she received an invitation to a reception at the White House.

"You know, we’re not best buds," Johnson said with a grin, characterizing their relationship as very respectful. "I think Tammy’s a nice person." Then, thinking about it some more, he added a caveat. "I would argue that the folks on the other side of the aisle, their ideology is destroying the country. But other than that, they’re nice folks."

Baldwin was more discreet. "We have a perfectly cordial relationship," she said in her soft-spoken murmur, which can be hard to hear.

The divides in Congress that have grown deeper in an era of increasing political polarization are on evident display in the words and actions of Wisconsin’s two senators. Johnson and Baldwin represent the kind of political diversity that has been relatively uncommon in Congress, a legacy of 19th-century politics when geography largely determined how people cast their votes for statewide office. Today, just 16 states have one Democratic senator and one Republican senator. Vermont has a Democratic senator and an independent, Bernard Sanders, who caucuses with the Democrats.

In several of those states, the senators often find areas of political agreement, or camaraderie at the very least.

Susan Collins, R-Maine, and her colleague Angus King, an independent who caucuses with the Democrats, have drawn up official letterhead with both of their names stripped across the top. A prolific texter, he sends her so many messages that her once-minimal texting skills now rival a teenager’s.

Mark Begich, a Democrat, and Lisa Murkowski, a Republican, regularly share long flights home to Alaska and live a block and a half from each other in Washington. They often chat when Murkowski is out walking her dog in the neighborhood.

Johnson and Baldwin have very little relationship inside or outside of the Senate. Neither could recall the last time they spoke at length.

Dinner together, perhaps? "Haven’t done that, no," Johnson said. "First of all, we don’t have time."

There was lunch once in the Senate dining room, which Baldwin said she brokered with the help of Rep. Jim Sensenbrenner, the senior Republican from Wisconsin. They had a businesslike time talking mostly about federal judges, but no one offered to make it a standing appointment.

Johnson won in 2010, a year with no presidential contest when energized conservatives were toppling Democratic candidates all over the country. It was a low-turnout election, with only 2.2 million Wisconsin voters showing up at the polls.

When Baldwin was elected in 2012, President Barack Obama was on the ballot, and three million people cast ballots for senator. The president carried the state, and she easily beat Tommy Thompson, a Republican former governor.

"This state has always vacillated from one extreme to another," said Rep. Ron Kind, a centrist Democrat from the western part of the state. "You go back to Bob LaFollette, the father of the modern-day Progressive Era, and then there was Joe McCarthy," he added, referring to two of the state’s most famous senators.

Kind said he was encouraged that Baldwin and Johnson have worked together when Wisconsin is concerned, citing their effort to fill empty seats on the federal bench. "That seemed to work well," he said. Besides, rivalries between members of the same party can often be far worse, others pointed out.

Those who have worked with both of them said their differences even extend to their delivery style and demeanor.

Johnson travels the state showing a PowerPoint presentation on the country’s financial woes — not exactly inspirational stuff, he is the first to acknowledge. But it animates him to the point that he can talk himself hoarse and now carries throat lozenges.

"He’s not trying to appeal to your heartstrings," said Robin Vos, the Republican speaker of the State Assembly. "He’s trying to appeal to your head. And I have to say, sometimes it’s a little depressing."

Baldwin’s low-key approach is what endears her to supporters. "She seemed to have an equanimity about her, almost a serenity," said Sen. Al Franken, D-Minn., one of her closest friends in the Senate, who started campaigning for her in the 1990s when she was running for the House.

Last month they had one rare area of agreement on a major piece of legislation that Republicans largely opposed: they both voted yes for the budget bill that will keep the government funded. And in a move that surprised many in his party, Johnson was highly critical of his Republican colleagues for forcing a government shutdown in October.

If Baldwin was at all heartened to hear him speak out against the shutdown, she wasn’t showing it. "I hope it would be an obvious fact that this was not something that was constructive," she said.

And Johnson emphasized that his budget vote and shutdown critique were not signs that he is going soft.

He likened his approach to government to taking candy away from someone with a sweet tooth — if candy were government-subsidized assistance, and the sweet tooths were the American people. "We are in the unenviable position of saying, ‘We know you like that candy. And we like it too,’" he said. "But if you notice, that candy caused you a cavity. It’s an abscess. It’s infecting the body. In the end it could kill you."

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Jeremy W. Peters, New York Times

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