The front-runner, Donald Trump, has already had to explain his past support for abortion and to deny he wants to raise taxes on the rich. Ben Carson, meanwhile, admitted on CNN's "State of the Union" last week he was once a "pretty left-wing Democrat."
Despite that liberal past, which would almost certainly doom any other Republican presidential hopeful, Trump and Carson are leading polls nationally and in the first-in-the-nation caucus state of Iowa. Their success is raising the question of whether the anti-establishment storm rocking the party means that ideological litmus tests set for candidates by conservatives on social and religious issues are less important than in the past.
Steve Scheffler, a Republican activist in Iowa, said the most important current factor driving the race is bitter resentment toward the Republican leadership in Washington.
"Activists are sick and tired of the go-along, buddy-buddy system, in particular John Boehner and Mitch McConnell and that whole political elite class that seems not to understand why they got elected," he said. "They are looking for areas to vent that frustration."
Such popular fury explains why Trump and Carson, who are not professional politicians but have passionately incited and exploited anger in the heartland, are doing well in the polls.
In the latest Monmouth University poll of GOP voters in Iowa released Monday, Trump and Carson were tied at 23%. Another outsider, former busineswoman Carly Fiorina, was third at 10%. They were leading the professional politicians, several of whom have strong conservative credentials, including Texas Sen. Ted Cruz, Wisconsin Gov. Scott Walker and Arkansas Gov. Mike Huckabee, a Baptist preacher who won Iowa in 2008.
Trump and Carson also lead the latest two national polls of the GOP nominating race, in a sign that past flirtations with the Democratic Party are yet to hurt them.
But five months before the first votes in the nominating race are cast in the Iowa caucuses, it is too early to write off more traditional political figures, especially those with solid conservative credentials.
If the anti-establishment fury were to ease, vulnerabilities and flip-flops by candidates with questionable conservative records could become more important and a candidate's positions on abortion, the defunding of Planned Parenthood, religious freedom, same-sex marriage and gay rights will come under scrutiny.
"Those issues are important to a bloc of voters that are essential to a Republican candidate's victory," said Ken Blackwell, a senior fellow at the Family Research Council and a former Ohio secretary of state. "The real question is not whether evangelicals are interested in Trump. Is he interested in them? He can't be if he is not interested in their core issues."
Blackwell warned that until Trump's true position on social issues becomes clear, "It's way too early for any social conservatives or Christian conservatives to endorse or summarily dismiss Mr. Trump or any other candidate."
Trump's ideological vulnerabilities exploded into the GOP primary race on Tuesday when Jeb Bush rebutted a barrage of attacks from the billionaire. The former Florida governor released a hard-hitting attack ad slamming the front-runner's conservative credentials, using damning footage of past statements.
"I probably identify more as a Democrat," Trump said, in archive film that also showed him backing Canadian-style state-run health care and praising Democratic front-runner Hillary Clinton as a "terrific woman."
Trump defended himself in an interview with CNN's Don Lemon on Tuesday, saying that in New York -- a liberal bastion reviled by conservatives -- "everybody was a Democrat."
He also predicted his background would prompt many voters who don't traditionally vote Republican to cross over and support the party in a general election.
"I think I will have a lot of Democrats voting for me -- far more than any Republican for the last long period of time," he said.
Carson said his current position on the major social issues of the day like abortion is clear.
"It is up to us, the people, to stand up for what we believe in, and what I believe in is life, and using the talent that God has provided to give life," said Carson at the annual National Right to Life Convention in July.
Trump has an even more explicit vulnerability on abortion. He said on NBC's "Meet the Press" in 1999 that he opposed a ban on partial birth abortion, though he hated the idea of the procedure, reasoning "I just believe in choice."
Trump now says he turned against abortion after seeing a friend decide not to terminate a pregnancy and have a child. He has long said he opposes same-sex marriage, even though he has gay friends.
Trump is also alarming some fiscal conservatives by raising the possibility of import tariffs on U.S. firms that manufacture abroad.
But in the CNN interview he dismissed the idea he would raise taxes on anyone other than on hedge fund managers, whose income is often taxed at lower rates than traditional wages.
"I'm not proposing to raise taxes," he said. I'm lowering taxes on the middle class and the upper middle class and the moderately wealthy. I said these hedge fund guys are making hundreds of millions of dollars and they are not paying tax."
Trump's attempt to close the deal with conservatives may ultimately depend on whether activists continue to overlook his past in a desire to embrace his wider anti-Democratic message.
That equation will play out especially in early voting states like Iowa, where evangelicals are crucial and ideological purity on social and religious issues is traditionally an important measure of Republican presidential contenders.
It may be a more problematic challenge for Trump than Carson. The former pediatric neurosurgeon has worked hard to build credibility on conservative issues on the talk radio circuit and elsewhere; he swept to national prominence by complaining about political correctness and the national debt to President Barack Obama's face at the National Prayer Breakfast in Washington in 2013.
Trump, meanwhile, may be thrown on the defensive for the first time in the campaign by Bush's ad, which implicitly argues that the former Florida governor and not the real estate magnate is the true conservative in the race.
"This is a pretty clean hit on Trump all wrapped up in an easy way for people to watch and share," said Craig Robinson, editor-in-chief of the "Iowa Republican" news website.
"While it might not help Jeb Bush much, I do think it starts a narrative where people will start to look back and say, 'Who is this guy who is leading the field?'"
Of course, there's precedent for candidates to move to the right -- and succeed.
As Bob Vander Plaats, president and CEO of the Iowa-based social conservative group The Family Leader pointed out, the patron saint of the conservative movement was once a devotee of liberal President Franklin Roosevelt.
"One of our very best presidents, Ronald Reagan, used to be a Democrat," he said.