One of the hot headlines of the week went something like this: Hillary Clinton called Republicans "terrorists." But the more important controversy is the actual policy fight behind the name-calling.

The gotcha version is mostly true: "Now, extreme views about women, we expect that from some of the terrorist groups," Clinton said on the stump Thursday. "But it’s a little hard to take coming from Republicans who want to be the president of the United States."

Understandably, Republicans have been eager to point out the use of the "T" word. The Republican National Committee called her language "a new low," and Ben Carson, a surging candidate in GOP polls, had this to say: "They tell you that there’s a war on women. There is no war on women. There may be a war on what’s inside of women, but there is no war on women in this country."

Provocative as Clinton's remarks were, though, they weren't really about terrorism. They were about a broader case that Republicans are trying to turn back the clock on a full set of women's health issues, from abortion to cancer screenings to the availability of contraception. So the highly charged rhetorical war between Clinton and Republicans offers a window into a real, substantive difference between the parties on the government's role in abortion and other issues under the umbrella of women's health.

"I would like these Republican candidates to look a mom in the eye who caught her breast cancer early because she was able to get a screening for cancer"

The question, posed by both Clinton and her Republican rivals, is whether it's more extreme for the government to subsidize abortion and Planned Parenthood, which uses its private funding for abortion services, or for the government to ban abortion and deny funding to groups like Planned Parenthood that are otherwise eligible for grants because they are engaged in providing abortion services.

This battle has tremendous implications for the 2016 presidential election, the outcome of which will make a big difference in how the US deals with abortion and funding for women's health clinics.

To understand the larger fight, you have to know all of what Clinton said

This isn't the first time a high-profile Democrat has drawn a parallel between Republicans and terrorists. Before and after the 2012 election, it was en vogue for top Democrats to say Tea Party Republicans acted like terrorists in legislative battles with President Barack Obama.

In those cases, the underlying point was that Tea Partiers were holding funding for the government and the nation's credit rating hostage to win concessions on a host of issues, including major spending cuts. In this case, Clinton was arguing that in their zeal to foreclose avenues to abortion, Republicans would hold hostage other important health-care service for women.

"I would like these Republican candidates to look a mom in the eye who caught her breast cancer early because she was able to get a screening for cancer, or the teenager who didn’t get pregnant because she had access to contraception, or anyone who’s ever been protected by an HIV test," she said.

The fundamental claim is that Republicans aren't respectful of women's access to health care — that they are treating women like second-class citizens. That, Clinton argued, is reminiscent of terrorist groups that repress and abuse women in their own societies. A Clinton campaign official denied that she had compared Republicans to terrorists, but the remarks were widely interpreted that way. And as she was reading from notes while she spoke, it's hard to make a credible case that Clinton didn't deliberately draw some kind of parallel between Republican presidential candidates and terrorists.

You can judge that for yourself from the video here and my transcription of her remarks immediately after it.

Marco Rubio brags about wanting to deny victims of rape and incest access to health care, to an abortion. Jeb Bush says Planned Parenthood shouldn’t get a penny. Your governor right here in Ohio banned state funding for some rape crisis centers because they sometimes refer women to other health facilities that do provide abortions. I would like these Republican candidates to look a mom in the eye who caught her breast cancer early because she was able to get a screening for cancer or the teenager who didn’t get pregnant because she had access to contraception or anyone who’s ever been protected by an HIV test. Now, extreme views about women, we expect that from some of the terrorist groups. We expect that from people who don’t want to live in the modern world. But it’s a little hard to take coming from Republicans who want to be the president of the United States, yet they espouse out-of-date and out-of-touch policies. They are dead wrong for 21st century America. We’re going forward, we’re not going back.

How the debate about abortion became entangled with cancer screenings, STD treatments, and contraception

Though restrictions on federal funding for abortion have long carried exemptions for victims of rape and incest, as well as expectant mothers whose lives are endangered by their pregnancies, the Republican Party platform in each of the last three presidential cycles has called for a constitutional amendment banning abortion without exception.

That's part of a marked GOP shift over several decades. Now candidates who hope to win the Republican nomination must be hard-line opponents of abortion. And even the definition of what constitutes an anti-abortion stance has become a moving target.

Since the Supreme Court concluded in 1973 that the Constitution protects a woman's right to have an abortion, anti-abortion activists have sought countless avenues to restrict abortion at the federal and state levels in ways that are consistent with the court's ruling.

The latest front in that war is an effort to cut off funding for women's health clinics that perform abortions with private money or that refer patients to abortion providers. For example, Oho Gov. John Kasich signed legislation denying money to rape crisis centers that refer patients to abortion providers, and fellow Republican presidential candidate Jeb Bush denied Florida state funds to Planned Parenthood when he was governor. Bush is campaigning on his promise to cut off federal subsidies for Planned Parenthood, a group that is one of the nation's most prominent and popular providers of health-care services for poor women.

The ground on abortion-ban exemptions is shifting, too

During the August 6 GOP debate, Rubio had a sharp exchange with Fox moderator Megyn Kelly over his position on whether legislation designed to curtail abortion, or federal funding for abortion, should carry exceptions in cases of rape or incest. Rubio has supported legislation with those exemptions — part of a long-codified triumvirate along with cases in which the expectant mother's life is in danger — but said he would also back legislation without them.

That was a surprise for many Rubio observers. He has supported the exceptions, including in legislation he has sponsored, because he wants to reduce the number of abortions. Given the current political alignment on abortion, bills that don't carry the exceptions have no chance of becoming law.

Democrats took immediate note of his position because they believe it can be used to portray him as insensitive to the wishes of women who have become pregnant through acts of sexual violence. Wisconsin Gov. Scott Walker has gone further, saying he doesn't believe in making exceptions in cases in which the life of the mother is endangered. Former Arkansas Gov. Mike Huckabee has said that abortion could be banned altogether under the Constitution if Congress passes a law giving "personhood" status to unborn babies.

Clinton and her allies think, as she said Thursday, that these are extreme positions that disregard women's rights to make choices about their own health — particularly when it comes to rape victims who need federal assistance to have abortions.

Cutting through the rhetoric, this is a consequential fight not only for abortion but for other women's health services

Republicans say Clinton went over the top by putting GOP candidates and terrorists in the same couple of sentences. Democrats say Republicans are waging a "war on women." And partisans on both sides of that divide will feel confident that their champions are right.

What's clear from the sometimes confusing rhetorical volleys, though, is that there is a very consequential debate being had over whether abortion is treated as a women's health issue and whether organizations that provide health care for poor women, including abortion, will have access to federal funding for non-abortion services.

The reason this has an economic element is that middle- and upper-class women have greater access to health care through private insurers, while women who depend on public subsidies often go to clinics, such as those operated around the country by Planned Parenthood, for a range of health-care services.

If federal funding for Planned Parenthood is cut off, it will certainly be harder for poor women to obtain both abortions and other health-care services. After all, Planned Parenthood is one of the nation's most prominent and most popular health care organizations.

A Reuters/Ipsos poll this month found that 54 percent supported federal funding for Planned Parenthood, while 26 percent opposed it. That despite the fact that, by a 44 percent to 34 percent spread, respondents who have seen the recent sting videos released by an anti-abortion group, which show Planned Parenthood executives discussing purchasing of tissue from aborted fetuses) reported having more negative views of the group afterward. (For the full rundown of this issue, read Sarah Kliff's explainer.)

And if Clinton and her allies win, poor women are sure to have equal or greater access both to abortion and to that set of health care services that Democrats say make the debate about a much larger philosophical divide between the parties over women's rights.