This week’s Gallup/USA Today poll looks like very bad news both for President Obama and the religious right.

Its most widely reported findings showed Mitt Romney erasing Obama’s once-formidable lead with female voters in swing states, one crucial to the president’s reelection hopes. But it also found that these voters care a lot about reproductive rights. Asked to name the most important issue for women in this election, almost 40 percent said abortion. Thirty-one percent of women said the candidates’ birth control policy would be “extremely important” in influencing their vote, and on that issue, respondents favored Obama by 21 points. These numbers suggested that Obama’s best chance of reopening the gender gap lay in emphasizing Romney’s support for hardline Republican positions on contraception and abortion.

It seems a safe bet that the campaigns are looking at internal polling saying the same thing, because suddenly both candidates are working hard to win over pro-abortion rights women. For Romney, who struggled to prove his anti-abortion rights bona fides during the Republican primary, this could be tricky. But social conservatives are likely to cut him quite a bit of slack, as a Romney victory could easily mean the end of Roe v. Wade. For that long-cherished dream, it’s worth putting up with a bit of rhetorical moderation.

At the debate on Tuesday night, Obama repeatedly brought up Romney’s pledge to eliminate federal funding for Planned Parenthood, while championing his own support for the federal mandate requiring insurance coverage of birth control. In the past, Romney has been outspoken against the mandate, calling it a violation of religious liberty, but on Tuesday, he tried to make it seem as if he backed it. “I don’t believe employers should tell someone whether they could have contraceptive care or not,” he said. “Every woman in America should have access to contraceptives. And—and the—and the president’s statement of my policy is completely and totally wrong.”

Then, on Wednesday, Romney’s campaign released a new ad aimed at moderate women. “You know, those ads saying Mitt Romney would ban all abortions and contraception seemed a bit extreme, so I looked into it,” says a young woman identified as Sarah Minto. “Turns out Romney doesn’t oppose contraception at all. In fact, he thinks abortion should be an option in cases of rape, incest, or to save a mother’s life.”

It isn’t the first time Romney has tried to distance himself from the absolutist anti-abortion position of his party—and his running mate. Last week, he told the Des Moines Register’s editorial board, “There’s no legislation with regards to abortion that I’m familiar with that would become part of my agenda.” That’s a pretty big turnaround from the stance he took during the primary campaign, when he published a statement titled “My Pro-Life Pledge” in National Review. In it, he promised, “[A]s president, I will support efforts to prohibit federal funding for any organization like Planned Parenthood, which primarily performs abortions or offers abortion-related services.” He also wrote that he would “advocate for and support a Pain-Capable Unborn Child Protection Act to protect unborn children who are capable of feeling pain from abortion.”

So far, Romney has been able to pivot away from such commitments because the anti-abortion rights movement, desperate to regain the White House, has been cooperative. Even before his spokeswoman walked back his comments to the Des Moines Register, leaders of the religious right were quick to absolve him. “No alarm bells here,” the Family Research Council’s Tony Perkins told Talking Points Memo. “I’m fine with his answer and he’s made it very clear he’s pro-life,” Ralph Reed told the conservative website NewsMax.

Such statements provided space for Romney to move even further toward the center, fudging his stance on the contraceptive mandate. Taken literally, his words during the debate don’t represent a reversal—Romney has never said that a woman’s employer should have the power to prevent her from getting birth control, just from using her health insurance to pay for it. In the context of his exchange with Obama, though, Romney’s words were clearly meant to obfuscate. He was certainly unwilling to make the type of religious liberty arguments that he once used before right-wing audiences.

For some conservative leaders, that’s just fine. “What he said was, ‘I don’t believe anyone should tell a woman whether she should have contraception or not, and that is completely consistent with his position on the mandate,’” says Marjorie Dannenfelser, president of the anti-abortion rights Susan B. Anthony List. “I don’t in any way fault him for trying to communicate what his position really is.”

Until Election Day, the right will likely keep giving Romney leeway, but that leeway will only go so far. He cannot, for example, renege on his promise to defund Planned Parenthood or to appoint Supreme Court justices who will overturn Roe v. Wade without sparking revolt. Indeed, one reason social conservatives are willing to tolerate their candidate’s current feint toward the center is because the possibility of finally getting rid of Roe is so tantalizingly close. Three of the Supreme Court justices who support the decision are now in their 70s. Romney only needs to replace one of them to usher in an era of total abortion bans in many states and perhaps at the federal level as well. As long as social conservatives know this, they don’t mind if Romney tries to make sure that most voters don’t.