WASHINGTON — Since his midterm election drubbing, President Obama has followed what seems like a wobbly course, taking one step to the left, then two to the right. The pattern, now reinforced by his tentative tax deal with Republicans, has pleased GOP adversaries while infuriating allies on the left.

But viewed through the lens of re-election politics, the strategy forged by the White House becomes clearer. The president is appealing to the independent voters who handed him one of the worst midterm setbacks in decades.

One of their major complaints has been Washington’s inability to work across party lines and get things done. And by trying to show that he can work with his nemeses, the president is laying a foundation for 2012.

“We have to get (the independents) back,” said a senior White House adviser, speaking on condition of anonymity while discussing internal strategy.

At the White House, Obama’s chief of staff, Pete Rouse, is finishing an internal review meant to prepare the president and his aides for the new political climate. It is expected to include a renewed effort to address one of the more surprising shortcomings of Obama’s first two years in office: his failure to forge a closer connection with everyday Americans.

“Since he stopped campaigning, he doesn’t seem to be reaching out and engaging the people anymore,” said Melanie Orpen, 38, an independent voter from suburban Philadelphia, during a focus group discussion Monday night sponsored by the University of Pennsylvania’s Annenberg Public Policy Center.

Whether the tax deal represents a new Washington reality or was simply a one-off response to an unusual problem, will be closely watched in coming months. Obama described the agreement as “a very unique circumstance.” And White House aides, in selling it, cautioned against predictions of “a new dawn in Washington,” as one of them put it.

The compromise offered goodies to the right, including a break on inheritance taxes and tax cuts for the wealthy, plus incentives for business. It also had something for those on the left, such as extended jobless benefits, expanded child tax credits for the poor and college tuition credits.

The administration has shifted from right to left among other issues. Recently, it reversed course on offshore drilling, imposing a ban on oil exploration off the Atlantic Coast and eastern Gulf of Mexico that seemed aimed at voters in the politically important states of Florida and Virginia.

That was followed by moves that pleased conservatives, such as a pay freeze on federal workers and a free-trade agreement with South Korea.

Obama’s post-election travels have taken him to two states he turned from Republican red to Democratic blue in 2008, Indiana and North Carolina. Issues he’s pushed on those trips, education and the government-aided revival of the U.S. auto industry, complement his plan to make jobs and the economy “my singular focus over the next year.”

Outrage on the left over what is seen as Obama’s capitulation on taxes for the rich only compounds a perception that he is not the man liberals thought they were voting for two years ago. An upcoming review of Afghanistan war strategy will do little to ease those concerns now that U.S. military forces, expanded by the president, are expected to remain into 2014.

But Obama’s political advisers express confidence that liberals will return to the fold, particularly as memories of actions taken 23 months before the election fade.

Thanks to his continuing popularity among Democratic voters, the president has room to maneuver. His standing with the Democratic base is stronger at this point than President Clinton’s was after the 1994 Republican takeover of Congress.

Unlike President Carter, there is no Ted Kennedy-esque figure preparing to challenge him in the primaries. A token opponent may yet surface, but Howard Dean, a liberal favorite, says he won’t run.

Also giving Obama cover is a new national poll that shows most Democrats don’t buy the liberal critique that Obama has caved in to Republicans.

A Pew Research Center poll completed last weekend, when a potential compromise was still developing, found that only about one-fourth of Democrats believe Obama is going along too much with Republican leaders. Independents wanted him to do still more; one-third said he was working with Republicans too little.

As part of Obama’s compromise with Republicans, the Bush tax cuts would expire after two more years. That puts the issue squarely into the middle of the 2012 presidential campaign, a prospect Obama says he welcomes.

The biggest threat to his re-election remains a sour economy, with unemployment near 10 percent and the prospect of high joblessness through 2012. For Obama to avoid the one-term fate of President George H. W. Bush, voters must sense that the economy has turned around.

His advisers say the tax compromise will inject fresh stimulus, boosting jobs and economic growth in 2011. Timing of an economic uptick could be crucial, notes Pew pollster Andrew Kohut, recalling that President Reagan’s re-election prospects improved as unemployment fell from double-digits to under 8 percent heading into the 1984 vote.

Obama needs results and must do a better job of selling them than he did in the first half of his term. Otherwise, the president will continue to look weak, a perception that is influencing public attitudes.

White House veterans say that after the midterm defeat, he may need time to find his footing. His State of the Union speech and 2012 budget offer opportunities to define the second half of his term, including, the president said Tuesday, a broad overhaul of the tax code.