WASHINGTON (Reuters) - For weeks, anxious Republicans called on Mitt Romney to respond more aggressively to attacks from the Democrats, afraid his hopes of winning the presidency could slip away if his rivals are able to define him as being out of touch with ordinary Americans.

This week, those Romney supporters are getting their wish.

Over the Internet and the airwaves, Romney and his allies have launched a stream of counterpunches against President Barack Obama and his team.

The effort is aimed at countering the Democrats' attempts to cast Romney as someone who shipped U.S. jobs overseas as a private equity executive at Bain Capital and hid much of his fortune in overseas accounts to avoid paying taxes.

The aggressive Republican strategy has included calling Obama a liar and demanding that a top Democratic Party official release her tax returns.

It is a shift from early July, when missteps by Romney's campaign on healthcare and immigration, among other issues, had some conservatives questioning his staff's competency and grumbling that Romney was playing too much defense while leaving Democratic attacks unanswered.

A speech by Romney on Wednesday, in which he was booed by a pro-Obama crowd at the National Association for the Advancement of Colored People (NAACP) convention, seemed to reflect that more aggressive strategy.

In what many analysts saw as Romney sending a signal to conservative white voters, Romney went before a not-so-supportive mostly African-American audience and criticized the first black U.S. president and his policies, particularly the healthcare overhaul the Republicans have dubbed "Obamacare."

At a fundraiser in Montana on Wednesday night, Romney told supporters: "I want people to know what I stand for and if I don't stand for what they want, go vote for someone else. ... Your friends who like Obamacare, you remind them of this: If they want more stuff from government, tell them to go vote for the other guy - more free stuff."

Romney occasionally has used the phrase "free stuff" to decry what he and other conservatives see as wasteful spending, although some Democrats see the phrase as a racially tinged reference to welfare.

'STEPPING UP THEIR GAME'

Republican consultant Matt Mackowiak said firing back at the Obama campaign's attacks suggests "the Romney camp realizes that they played defense all last week and they are stepping up their game."

Opinion polls indicated that Democratic attacks on Romney's record at Bain helped Obama gain ground with many Americans, notably in politically divided industrial states such as Ohio and Pennsylvania, both battlegrounds in the November 6 election.

Some of Obama's claims about Romney's role in outsourcing jobs at Bain "have hurt Romney, particularly in the Rust Belt," Mackowiak said.

Romney's team argues that an Obama ad calling Romney an outsourcer of jobs was unfair and cites various political fact-checking columns that said as much.

A new ad from Romney's campaign, addressing Obama's claims about the Republican's business record, calls the president a liar running a "dishonest campaign." It is also firing back at Obama by accusing the president of being "the outsourcer-in-chief" and using "taxpayer dollars to create jobs overseas."

The attacks focus on job-creation programs in Obama's $800 billion economic stimulus program, and are essentially based on an assumption that any money sent to a foreign-owned company goes to hire foreign, not American, workers.

Analysts have called that leap in logic a stretch.

"It seems rather strange for Republicans, who claim to be defenders of free enterprise, to be making an argument that foreign companies should not receive federal money even if it is used to hire American workers," The Washington Post's political fact-checking column said on Thursday.

ROMNEY'S TAX RETURNS

The new wave of attacks by Romney's camp also fights back against Democrats' calls for the former Massachusetts governor, who has a fortune of up to $250 million, to release more of his tax returns.

In January, Romney released his family's 2010 return and an estimate of its filing for 2011. They indicated that Romney is holding millions of dollars in offshore accounts and once had a Swiss bank account.

The disclosures raised questions about the lengths to which Romney has gone to shield his money from taxes - and inspired Obama campaign's suggestions that Romney is hiding something.

Republicans have countered that Obama is diminishing himself with personal attacks targeting capitalism, and are demanding the release of Democratic National Committee chairwoman Debbie Wasserman Schultz's tax returns.

Wasserman Schultz has been a leading voice in the tax offensive against Romney, but it is otherwise unclear why she has been targeted.

Since 1968, most presidential candidates - including Romney's father, George, who sought the Republican nomination that year - have released years of tax records. Members of Congress, including Wasserman Schultz, generally do not. She has responded to Republicans' calls for her to release her returns by noting that she is not running for president.

Mackowiak, the Republican strategist, said there was some risk Romney's strategy of striking back at Obama could alienate independent voters being courted by both sides. Conservatives may have been thirsting for Romney to hit back harder against the Democrats, but he also will need to win over independents to oust Obama from the White House in the November 6 election.

"Base Republicans unquestionably want to see Romney fight," Mackowiak said. "The question is, what happens to swing voters if both sides are throwing haymakers for four months?"