MOUNT VERNON, Iowa (Reuters) - President Barack Obama hit rival Mitt Romney hard on women’s issues as he headed back on the campaign trail on Wednesday after a spirited debate performance that re-energized his bid for a second term.

A day after a much-improved performance in the second of three presidential debates, a revitalized Obama continued sparring with his Republican opponent, making fun of Romney’s comment that he had received “binders full of women” to consider for cabinet positions when he was governor of Massachusetts.

“I’ve got to tell you, we don’t have to collect a bunch of binders to find qualified, talented driven young women ready to learn and teach in (science, technology and engineering) right now. And when young women graduate, they should get equal pay for equal work,” Obama, relaxed and smiling in shirt sleeves and a loosened tie, told 2,000 people at Cornell College in Iowa.

With 20 days to go until the election, Obama campaigned in Iowa and Ohio while Romney was in Virginia - all important “swing states” that can go to either candidate on November 6.

In Chesapeake, Virginia, Romney said Obama has failed to help women get well-paying jobs and also accused the president of failing to produce a second-term agenda.

“Don’t you think it’s time for them to finally put together a vision for what he’d do in the next four years if he were re-elected?” Romney asked about 3,500 supporters outside a community college.

Romney scored points of his own during Tuesday night’s debate when he focused on middle class economic struggles and listed promises he said Obama failed to keep from his 2008 campaign.

Both sides claimed victory, but most polls gave a badly needed edge to Obama, who saw his lead in polls contract sharply after a lackluster performance in the first debate October 3.

Voters said Obama outperformed Romney by a substantial margin on Tuesday night, according to a post-debate Reuters/Ipsos survey: 48 percent to 33 percent.

“This will give the president a bit of a bounce and a little bit of an edge, but it’s going to be quite close right down to the wire,” Notre Dame University political science professor Michael Desch said.

The final presidential debate is scheduled for Monday in Boca Raton, Florida.

OBAMA HOLDING SLIM POLL LEAD

Obama leads Romney by 47 percent to 44 percent among likely voters, according to Wednesday’s Reuters/Ipsos daily online tracking poll. His 3-point lead was unchanged from Tuesday, with most of the interviews done before the latest debate.

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A Rasmussen Reports tracking poll of 11 swing states had Obama leading Romney by 50 percent to 47 percent on Wednesday.

Obama needs strong support from women voters if he hopes to beat the Republican, and he made sure to appeal to them during the debate by bringing up contraceptive rights and his push to ensure pay equity.

Analysts said Obama did particularly well on women’s issues, boosted by Romney’s awkward “binders” statement, which lit up social media. The mock Twitter account @RomneyBinders amassed more than 33,000 followers, and a Facebook page “Binders Full of Women” attracted more than 303,000 “likes.”

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Romney, a former private equity adviser, hit back by contending his business experience will help women, and all Americans, by bolstering the sputtering economy.

His campaign also released new television advertisements directed at women.

One outlines Romney’s stance on abortion and contraception, which is more moderate than that of many Republicans. In the ad, a woman directly faces the camera and talks about Romney’s support for contraception as well as abortion in cases of rape, incest or a threat to a mother’s life.

A second, called “Humanity,” features women who worked for Romney when he was governor of Massachusetts talking about his sensitivity to women employees.

Analysts also said Romney bungled on foreign policy when he mischaracterized - and was corrected by the debate monitor - Obama’s initial remarks on last month’s deadly attacks on diplomatic facilities in Libya.

Obama took advantage of the moment to accuse Romney of politicizing the deaths of four Americans.

Polls show the economy is an area in which voters view the two candidates similarly, or give the Republican an edge. But Obama has been helped recently by some positive economic news.

On Wednesday, the Commerce Department said groundbreaking on new homes surged in September to its fastest pace in more than four years, a sign the sector’s budding recovery is gaining traction and supporting the wider economic recovery.