Gallup followed up on CNN’s weekend poll with a survey of its own, taken among general population adults on Monday rather than the weekend, but found much the same result.  Overall, the budget deal gets strong support, 62/25, with majorities of Democrats, Republicans, and independents supporting the plan.  A large majority wasn’t so generous with assigning credit for the result, however:

Six in 10 Americans approve of the 11^th-hour federal budget agreement that congressional leaders reached in time to avert a government shutdown. Support for the deal made on Friday is somewhat higher among Democrats than among independents and Republicans, 71% vs. 60% and 58%, respectively.

Few Americans see a political winner in the outcome — with 5% saying it was a victory specifically for the Democrats, 8% specifically for the Republicans, and 20% for both. Rather, the majority of Americans, 56%, say the long-negotiated compromise was not a victory for either side.

Republicans are more likely than Democrats to believe their own party was victorious — 16% vs. 6% — however, the majority of both groups believe neither side won.

That’s the nature of compromise.  Usually, neither side feels it has won as it mulls over what had to be sacrificed.  In this case, though, the issue may fall onto the numbers themselves.  This battle was just the opening round, a chance to set the parameters of the bigger debates on entitlement reform and other structural issues that drive the stakes far higher than what was on the table in this round of bargaining — a mere 6% of the overall budget at $250 billion or so.

The stakes get higher in Round 2, and the challenges do as well for Republicans.  According to the survey, raising taxes on households with income over $250,000 enjoys majority support, 59/37.  Independents support these higher taxes for wealthier Americans 60/37.  Republicans may have to find a way to give on that issue or better explain why it will hurt the economy and result in no real revenue increase.

Fortunately, the debate in Round 2 will be shaped by what happened in Round 1.  As I write in my column for The Week, we got a look at which national leaders are most consequential, and regardless of the specifics of the deal, Democratic leadership came off poorly and on the defensive:

Just a few weeks ago, Democratic leadership in the Senate proclaimed that anything more than $10 billion in total cuts to the FY2011 budget proposal was too radical to be tolerated. Three weeks ago, Sen. Chuck Schumer (D-N.Y.) called them “extreme.” By Saturday, Senate Majority Leader Harry Reid (D-Nev.) bragged about the $39 billion in cuts as “historic.”

Speaking of Reid, where was he in Obama’s estimation of power in Washington? Exactly nowhere. Reid turned into a potted plant in the budget negotiations. Boehner dealt directly with the White House to get what he could and cut the deal. Reid could have forced his caucus to pass a realistic budget earlier in the year, which would have moved the debate to a conference committee and kept Obama out of it. Instead, Reid’s failure made him irrelevant. Boehner can go directly to the Oval Office in the future, making budget battles bilateral and minimizing the necessity of trading more away than he needs to reform federal spending.

That’s already had an impact:

Until now, Democrats have either scoffed at entitlement reform entirely or considered it a revenue rather than a spending problem.

Not anymore. This week, President Obama will roll out his response to the Ryan plan instead of staying on the sidelines, as he has for much of this year. His plan is expected to call for tax increases, notably the same increase on those earning $250,000 or more that Obama agreed in last December’s tax compromise to postpone until 2013. However, the White House signaled this weekend that it will also include actual and significant cuts in entitlement programs, an acknowledgment that the terms of the debate have changed, and that Republican efforts to demonstrate fiscal leadership have the White House worried about their prospects in 2012.

But the Gallup results show that Obama’s approach to call for both tax hikes and entitlement cuts will resonate as a reasonable compromise unless Republicans continue to show leadership on the federal budget.  It’s a real danger of overplaying the hand, especially now that Obama has suddenly re-embraced his own deficit commission.

Update: Jen Rubin notes the same dynamic in the Beltway:

President Obama has promised to speak about his thoughts on the debt crisis. It’s not clear, however, why he didn’t do this at the State of the Union or what, if anything, he is going to accomplish. But as a matter of communications strategy, there’s reason to question sending the president out to talk to the country in these circumstances. An adviser of a senior Senate Republican has this take: “They didn’t think this one through. They’re winging it.” He sees a three-pronged dilemma for the president: “His base won’t let him touch Social Security, Medicare was gutted in ObamaCare, and they couldn’t pass a tax hike with a supermajority Democratic Congress.”

A Republican communications guru also takes a dim view of the effort, telling me, “This speech, to me, is incredibly reactionary, as is everything they seem to do at the White House. Paul Ryan made a big splash with his plan, and now the White House is playing catch-up. Notice that the speech is in middle of the day and not at the White House but instead at George Washington University. So, it’s a ‘major-minor’ speech?” The guru sees a White House obsessed with spin: “All the White House believes the president has to do with this speech is reclaim the headlines. So, he just has to sound good. In their mind, he could be reading out of the phone book.”