The dome of the U.S. Capitol Building is seen as the sun sets on Capitol Hill in Washington on March 7. Committees in the House and the Senate have launched a bipartisan campaign to lay the groundwork for a rewrite of the tax code. (Charles Dharapak/AP)

Dueling budgets and three days of political schmoozing by President Obama left Congress on Thursday in broad agreement on at least one idea: The tax code is awful and should be overhauled. But actually fixing it faces long odds given deep divisions between the two parties on the role of taxes in the economy.

Members of the Senate Finance Committee from both parties met for the first time Thursday to start laying the groundwork for legislation to overhaul the tax code, as the House Ways and Means Committee engages in a similar bipartisan effort.

And while House Republicans and Senate Democrats this week advanced budget plans that are at sharp odds over most policies, they agreed that the economy would benefit from wiping out tens of billions of dollars a year in existing tax breaks.

Where they differ, however, is over what to do with the savings — and that dispute could snuff out any hope for the grand bargain Obama pursued in separate meetings with each party in both chambers this week.

Republicans see the tax code as another example of big government, with dozens of tax breaks that favor certain industries and distort the economy. They want to eliminate most of those tax breaks and return all the money generated to taxpayers through lower rates. Nobody would pay higher tax rates, and Republicans believe that a simpler tax code would unleash economic growth.

Democrats see the tax code, however, as riddled with giveaways to the rich and want to scale back deductions and loopholes to raise taxes on the wealthy by $1 trillion. They see those tax breaks as spending programs that subsidize the lives of wealthy Americans at a time of spending cutbacks on programs that help the poor and middle class.

“The starkest difference is that Republicans are pursuing fundamental reform to grow the economy and make the U.S. far more competitive around the world,” said Rep. Kevin Brady (R-Tex.), chairman of the Joint Economic Committee. “While some Democrats view it the same way, many are seeing this as an opportunity for higher tax revenue or higher tax rates.”

But the top Democrat on the House Budget Committee, Chris Van Hollen (Md.), said closing tax breaks and using that revenue to pay down the deficit is critical.

“I think if we want to reduce our deficit in a steady, responsible way, we need to do it with some additional revenue,” he said. “If you don’t have any additional revenue, then you’re reducing the deficit in ways that damage other important national priorities like our investment in education, like our investment in science and research.”

Van Hollen added that without one side ceding ground, no broad agreement to tame federal borrowing or even turn off the sequester will be possibe. “This is absolutely key to getting an agreement,” he said. “This is the key that can help unlock the door.”

Obama concluded three days of visits to Capitol Hill on Thursday, but White House officials downplayed expectations for a deal anytime soon.

“There remains enormous obstacles,” White House press secretary Jay Carney said Thursday. “There remains at least in some corners of the Republican Party an absolutist position that says, ‘No way, no how.’ ”

This year, individuals and companies are expected to pay $1.3 trillion less in taxes as a result of deductions, loopholes and special rates — including breaks as varied as the deduction for mortgage interest, the exclusion for employer-sponsored health insurance and special rates on investment income.

For the most part, the two sides agree that the tax code creates enormous complications and that it would be better to be simpler. While many of the best-known provisions, such as the deduction for mortgage interest, have widespread political support, economists on both sides of the spectrum question their value.

For instance, many economists agree that though popular, the mortgage interest deduction does little to encourage homeownership, while the health insurance exclusion pushes up health insurance costs. Democratic-leaning economists and politicians also complain that tax breaks tend to favor the wealthy, while Republicans say that some are especially justified — for example, low rates for investment activities.

In his meetings with lawmakers this week, Obama spoke in favor of tax reform as an engine for fresh revenue, but he and other Democrats have been less interested in helping Republicans achieve their primary goal — lower tax rates. Especially in the wake of the “fiscal cliff” deal that raised the top rate from 35 percent to 39.6 percent, many Democrats are more inclined to pocket the extra money than to embark on a difficult and politically perilous tax rewrite.

Senate Finance Committee Chairman Max Baucus (D-Mont.) and House Ways and Means Committee Chairman Dave Camp (R-Mich.) are nonetheless forging ahead with bipartisan discussions. But the success of their efforts depends heavily on the outcome of negotiations over a broader deficit-reduction deal that resolves the question of whether tax reform raises cash or lowers rates.

Those negotiations are likely to take months, but leaders of both parties see the adoption this week of competing budget resolutions in the House and Senate as a first step. The House Budget Committee approved a plan late Wednesday on a party-line vote to balance the budget over the next decade entirely through spending cuts, and it will be debated on the House floor next week.

The blueprint, drafted by House Budget Committee Chairman Paul Ryan (R-Wisc.) with Camp’s help, calls for “revenue neutral” tax reform that lowers the top rate from 39.6 percent to 25 percent — a goal Democrats say cannot be met without shifting more of the tax burden onto the middle class.

The Senate Budget Committee was set to approve its own partisan budget late Thursday that seeks to replace automatic spending cuts known as the sequester with $1.8 trillion in alternate policies, including nearly $1 trillion in new taxes over the next decade.

The Senate Democratic blueprint orders the Senate Finance Committee to raise the cash through tax reform by Oct. 1 — a mandate that Republicans say would make a bipartisan tax reform bill impossible. The full Senate is also expected to take up its budget next week.

Assuming both chambers adopt their resolutions, party leaders are expected to appoint a conference committee — headed by House Budget Committee Chairman Paul Ryan (R-Wisc.) and Senate Budget Committee Chairman Patty Murray (D-Wash.) — that could become the primary forum for continuing discussions over a big deal.