If Carly Fiorina gains any traction from her barbed attacks on Hillary Clinton, the right-wing cartoons will practically draw themselves: Carly and Hillary in a teeth-baring cat fight, Carly’s claws like a tiger’s, HRC’s eyes as red as a Demon Sheep’s, their hair seriously mussed, and Benghazi burning in the background.

As one man tweeted, “Let the Cat Fight begin!! Fiorina will tear Hillary to shreds.”

“Fiorina vs Hillary in 2016,” someone else raved. Why? “Because men love a cat fight.”

It is indeed a male dream, especially males who are Republican presidential candidates (and who isn’t?). If Carly handles the edgy, personal attacks on Hillary, they figure, we won’t get Rick Lazio-ed off the stage.

But at the press conference-ambush that Fiorina held outside a South Carolina hotel where Clinton was speaking, the former Hewlett-Packard CEO bristled at suggestions that she was doing the male Republicans’ dirty work. Fiorina, Maggie Haberman wrote,

quickly grew discomfited when the questions seemed to treat her more as a heckler pulling a stunt than as a formidable candidate making an otherwise significant campaign stop.

One reporter asked if Ms. Fiorina was being used by the men in the Republican field to harass Mrs. Clinton.

Ms. Fiorina insisted she had planned her trip here “many, many weeks ago, so perhaps she’s following me.”

The first female CEO of a Fortune 20 company has had to deny that she’s a tool of the GOP boys—the poisoned-tip of their spear—for a while now. “The party is not leaning on me to do anything, and I didn’t ask the party’s permission,” she said in March.

She’s her own woman, independent, thinks for herself. But in her self-appointed role as Hillary’s foil, there’s a fascinating tension between the politics of cat-fighting and her feminist-tinged complaints about just those sort of stereotypes. “I think the media hold women to different standards,” she said at the same press conference. “[The press] scrutinizes women differently, criticizes women different, caricatures women differently.”

But even as Fiorina wants everyone to know that’s she’s more than Hillary’s would-be bête noir (“the vast majority of my speeches in front of anyone are about a host of issues,” she told reporters), in many ways she is also Hillary’s doppelganger. They have much in common:

1) Most obviously, both are women vying for power in a man’s world. They are considered “demographically symbolic” (as the NRA’s Wayne LaPierre sneered about Obama’s entire presidency), and they each have a deck of “gender cards” to draw from. It’s perfectly legit, since women have been largely banned from the card games. But while Hillary is playing pinochle, Fiorina is playing something closer to three-card monte. As Amanda Marcotte writes:

In recent months, Fiorina has shown that the only thing she loves more than deriding those who play the “gender card” is playing the gender card….

“If Hillary Clinton were to face a female nominee, there are a whole set of things that she won’t be able to talk about,” Fiorina told reporters in April. “She won’t be able to talk about being the first woman president. She won’t be able to talk about a war on women without being challenged. She won’t be able to play the gender card.” No she won’t, because I, Carly Fiorina, will play it for her!

2) They are so much alike that Fiorina claims Hillary stole the title of her recent memoir, “Hard Choices,” from Fiorina’s account of her widely criticized tenure at HP, “Tough Choices,” which was published in 2006.

“And last month,” Amy Chozick writes, “after Mrs. Clinton urged 5,000 female tech professionals in Silicon Valley to ‘unlock our full potential,’ Ms. Fiorina again accused Mrs. Clinton of stealing: Her leadership political action committee, an aide to Ms. Fiorina noted, is called the Unlocking Potential Project.”

3) That could be a case of bad-faith borrowing, or it could all be a coincidence. As Jason Linkins points out, “’unlock your potential’ may actually be the most banal phrase these Thought Leader types employ.” Fiorina seems to think she owns this sort of MEGO boilerplate because she comes from big biz, but the truth is that both women are corporatist, business-friendly politicians, supported by industry and Wall Street.

4) Both women have worked very hard to please powerful peers who are almost without exception white males—at HP, Fiorina laid off more than 30,000 people (see CarlyFiorina.org); and Hillary voted for the Iraq War as she was preparing to run in 2008.

Fiorina has to be aggressive, of course, because she’s the newbie—never won an election, while Clinton was a two-term senator from New York, not to mention secretary of state. But one interesting difference between the two is that Hillary has refrained from attacking other women politicians—even, in 2008, Sarah Palin. “You know, I don’t want to be the chick police,” says Nicole Wallace, who famously quit the Palin campaign, but Fiorina’s focus on Hillary, she says, “runs the risk of having it look personal.”

It wouldn’t be Fiorina’s first foray into the too-personal: when she thought she was off mic during her 2010 race against California senator Barbara Boxer, Fiorina said she had seen her opponent on TV and wondered, “God, what is that hair? So yesterday.”

Naturally, the Democratic Clinton and the Republican Fiorina differ on issue after issue—immigration reform, abortion, equal pay, foreign policy. But let me acknowledge that in writing about these two women, I have, like so much of the media, completely stayed away from any mention of any issue.

Because… CAT FIGHT!