We already know the outlines of the biggest fights to come in Washington, D.C.’s new divided government — showdowns over health care, spending and the debt limit are sure to dominate. But divided government may also create opportunities for big compromises.

A Democratic Congress and President Ronald Reagan cut deals on Social Security, tax reform and immigration. President George H.W. Bush made a major deficit-cutting deal with Congressional Democrats in 1990. And a Republican majority cut major deals with President Bill Clinton on welfare reform in 1995 and the budget deficit in 1997.

Already, senior leaders of both parties are eyeing areas where they see compromise as possible.

Senate Minority Leader Mitch McConnell, despite telling the Heritage Foundation that his top goal is defeating President Barack Obama in 2012, has pointed to a host of issues that he said are ripe for bipartisan legislation — including trade agreements, nuclear power, electric cars, clean coal, entitlement reform and tax reform. The Kentucky Republican has already scored the biggest deal since the elections: the two-year extension of all the Bush-era tax cuts during the lame duck.

Senate Majority Leader Harry Reid has also talked hopefully of making bipartisan progress, although without much detail, and chairmen and ranking members on various committees have started talking about their agendas for the year.

The Nevada Democrat has talked up the possibility of doing something on energy, which could come together as gas prices head upward again. Senate Finance Chairman Max Baucus (D-Mont.) has started looking at a possible tax reform package.

Senate Judiciary Chairman Patrick Leahy, meanwhile, said recently he is looking for areas where he and new ranking member Chuck Grassley (R-Iowa) can write bipartisan bills, including on the long-stalled patent reform bill, fraud and cybersecurity.

“I know that we can be productive,” the Vermont Democrat said. “I worked with a Republican House chairman to enact the first Justice Department authorization in 25 years. I worked with a Republican Senate majority and Republican president to extend the Voting Rights Act and to enact the Innocence Protection Act, which provides post-conviction DNA testing for those wrongfully convicted.”

Those are the kind of under-the-radar issues that might be able to get across the finish line while the parties duke it out over the GOP’s attempts to repeal last year’s health care reform bill and to slash domestic discretionary spending by $100 billion a year, aides said.

“There are these high-stakes [issues] and that’s what everybody focuses on, but there are these issues like patent reform, which are contentious but aren’t partisanly contentious, that could get done,” one senior Republican aide said.

Some compromises will have to come quickly to avoid potentially catastrophic consequences. A continuing resolution funding the government expires March 4, and Treasury Secretary Timothy Geithner has warned Congress of disaster if the $14.3 trillion debt limit isn’t raised by the end of March.

Republicans are demanding deep, unspecified spending cuts but have yet to name which programs they want to cut, and they are looking to see what Obama offers up in his State of the Union address and his fiscal 2012 budget, which is due in the next few weeks. The two sides may find themselves in a high-stakes game of chicken, with the specter of a government shutdown or an unprecedented default on the federal debt if they cannot make a deal.

Another area to watch is whether Republicans and Democrats find common ground on individual appropriations bills, or whether those measures will be derailed by partisan squabbles over spending levels, earmarks and social policy riders.

Republicans have adopted an earmark moratorium for the 112th Congress, but no such ban exists in the Senate. House Republicans have also vowed to defund the health care law, which could also lead to a showdown with the Senate and the president.

To pass any spending bills this year, House Appropriations Chairman Hal Rogers (R-Ky.) will have to corral votes for bills that will be less about doling out largess and more about taking an ax to federal spending.

Appropriations ranking member Norm Dicks said he was cautiously optimistic that appropriations could be one area where deal-making flourishes, at least on process.

“We’re hoping we can go back to regular order, which is what the Republicans want,” the Washington Democrat said. “I certainly favor that, where we have subcommittee markups, full committee markups, go to the floor, open rules.”

Democrats and Republicans have sparred in recent years over Democratic attempts to limit amendments to appropriations bills, which traditionally had an open amendment process.

“I hope we can do that,” Dicks said. “What we’re going to have to do is have cooperation from the Members. In the past, once a bill was on the floor, the chairman and the ranking member would get together and work out an agreement … on amendments.”

“I think our side is really willing to cooperate on this,” Dicks said, noting that top House Democrats have sat on the Appropriations panel.

As part of his effort to overhaul House operations to give the minority more input, Speaker John Boehner (R-Ohio) has promised that “many bills,” including spending bills, will be brought to the floor under an open rule.

Blue Dog Democrats, meanwhile, hope to play a role as a bridge between the White House, House Republicans and the Senate. Their leaders told Roll Call they are looking for bipartisan accomplishments like the 1995 welfare reform law.

House Budget ranking member Chris Van Hollen said Democrats will look for areas of common ground with Republicans “where they exist.”

“We’ll have to see, going forward, where we can find common ground,” the Maryland Democrat said. “Where we disagree, we need to express those differences respectfully.”

Van Hollen and his counterpart, Budget Chairman Paul Ryan (R-Wis.), already have locked horns, including over a Republican move to empower Ryan to effectively deem a budget number on his own, which Van Hollen ripped as “an unprecedented grant of authority … to decide something unilaterally without debate or a vote of the House.”