Is there a “war on women” going on? That is a matter of dispute between the parties these days.

But one thing is certain: There is a battle raging over them.

If that wasn’t clear after weeks of argument over contraceptive coverage, it became so Thursday, when caterpillars and country clubs got dragged into the fray.

In an interview with Bloomberg Television, Republican National Committee Chairman Reince Priebus dismissed the “war on women” meme as a concoction of Democrats and their sympathizers in the news media.

“If the Democrats said we had a war on caterpillars, and mainstream media outlet[s] talked about the fact that Republicans have a war on caterpillars, then we have problems with caterpillars,” Priebus said.

No surprise, that brought a swift reaction from Democrats.

President Obama’s deputy campaign manager, Stephanie Cutter, issued a statement contending that the RNC chairman’s “comparison of Republican attempts to limit women’s access to mammograms, cervical cancer screenings, and contraception to a ‘war on caterpillars’ shows how little regard leading Republicans, including Mitt Romney, have for women’s health.”

Also on Thursday, Obama weighed in on whether women should be admitted as members to the all-male Augusta National Country Club, site of this week’s Masters tournament. The long-standing dispute has gained currency this year because a woman, Virginia Rometty, is now chief executive of IBM, a longtime sponsor of the tournament whose previous chief executives have been admitted to Augusta.

The president’s “personal opinion is that women should be admitted,” White House press secretary Jay Carney said, adding that Obama thinks “it’s long past the time when women should be excluded from anything.”

Two Republican presidential candidates followed with their own declarations that the club should admit women.

“I’m not a member of Augusta. I don’t know that I would qualify — my golf game is not that good — but certainly if I were a member and if I could run Augusta . . . of course I’d have women in Augusta,” Romney said while campaigning in Pennsylvania.

And former House speaker Newt Gingrich declared via Twitter that he would like to see his wife admitted. “I think callista would be a great member #Augusta -maybe she would let me come and play,” he tweeted.

What voters should object to, many Republicans say, is Democrats’ efforts to characterize political disputes as a “war on women.”

“I find it offensive that the Democratic National Committee is using a term like that to describe policy differences,” said Sean Spicer, a spokesman for the Republican National Committee. “It’s not only bad, but it’s downright pathetic they would use a term like ‘war’ when there are millions of Americans who actually have engaged in a real war. To use a term like that borders on unpatriotic.”

The RNC responded to Obama’s comments about Augusta by mass e-mailing an article noting that the president has been criticized in the past for playing golf almost exclusively with men. The headline of the story, published by the Washington Free Beacon and linked by the Drudge Report, is “Obama’s War on Women.”

In an election that was supposed to be all about the economy, gender politics have been advancing to the forefront. Thanks to a series of recent GOP missteps, Democrats see an opening to ensure and expand their traditional advantage with female voters.

A USA Today-Gallup poll of a dozen battleground states released this week underscored how fluid their allegiances are this election season.

Among independent women — a key group of swing voters — Obama had been trailing Romney by five points in a series of surveys late last year. But that number shifted dramatically in polling conducted in February and March, and the president took a 14-point lead over the former Massachusetts governor, marking a net gain of 19 points.

Independent men shifted in Obama’s direction as well, but in smaller numbers. Obama trailed Romney by 11 points late last year among independent men in the USA Today-Gallup poll, but he moved into a one-point lead in February and March, a shift of 12 points.

Pollsters in both parties said Romney’s hopes for victory will hinge in part on narrowing the gender gap.

“We will have some strategies of dealing with it,” said GOP pollster Linda DiVall. “We will have to if we want to be successful in November.”

Neil Newhouse, who is the lead pollster for Romney’s campaign, said the candidate’s problems with women probably represent collateral damage from the arguments that women have been hearing about contraception and other social issues, rather than any reflection on Romney’s positions.

“There’s nothing I’ve seen that suggests this is Mitt Romney-specific,” he said.

However, it doesn’t help that Romney appears to rely on his wife, Ann, to translate the concerns of 51 percent of the population.

Asked about the latest polling on Thursday, the former Massachusetts governor said, “My wife has the occasion, as you know, to campaign on her own and also with me, and she reports to me regularly that the issue women care about most is the economy.”

Ann Romney, along with the wives of two of her husband’s rivals, Callista Gingrich and Karen Santorum, will headline a luncheon next week with women of the National Rifle Association.

Obama, meanwhile, is losing few opportunities to show his feminist side. On Monday, he raved to CBS Sports about his love of coaching his daughter Sasha’s youth basketball team.

During halftime of the NCAA men’s college basketball championship game, Obama told analyst Clark Kellogg that he gets more joy out of that than he felt when he played.

“I bleed when those girls play,” said the president, who also noted that girls and women have far more athletic opportunities than when he was growing up.

Last month, the Obama campaign sent mailers promoting the administration’s achievements for women to hundreds of thousands of female voters in swing states.

On Friday, the White House will host a conference on women and the economy, during which officials will “highlight what we have accomplished together and the challenges that remain,” a senior aide said Thursday.

In a background briefing on the conference for reporters, administration officials said the budget plan passed by the Republican House would disproportionately harm women with its proposed cuts to Medicare, Medicaid and Social Security. The argument goes that because women generally live longer than men, they rely more on those programs.

And in his stump speeches, Obama often notes that the first bill he signed into law, the Lilly Ledbetter Fair Pay Act, gave women greater leeway in filing suit to receive equal pay for equal work.