Correction: Earlier versions of this article, including in the print edition of Monday’s Washington Post, incorrectly identified the Bloomberg reporter who interviewed former Florida governor Jeb Bush about the 2012 GOP nominating contest. It was Mark Silva.

The startling outcome of the South Carolina primary did more than turn the race for the Republican presidential nomination upside down. It also left GOP voters with a choice that only serves to highlight the party’s year-long identity crisis and search for a standard-bearer.

If the voters have seemed puzzled, it is no wonder. The field of candidates they have examined over the past months has not included anyone who yet seems to possess the attributes that can provide a center of gravity. Now, seemingly, it is down to Mitt Romney and Newt Gingrich, but can either deliver what the party needs?

In some respects, the contest between Romney and Gingrich falls into a familiar construct of Establishment vs. Insurgents, and yet neither candidate is the ideal to play his assigned role, or, more important, to bridge the divides within the party that characterized many primary contests in 2010.

Romney has the breeding and coun­tenance of the establishment, but when he first ran for office he ran away from Ronald Reagan. As a registered independent, he voted for Paul Tsongas in the 1992 Democratic presidential primary. He can seem out of place in a party whose conservative Southern base has been its core personality since the mid-1990s.

Gingrich has captured the anger of the grass roots and channeled the deep dislike of President Obama that gave rise to the tea party. But like Romney, he has limitations playing the role of insurgent, although his rhetoric has long been that of a bombastic outsider.

He has operated in the corridors of power in Washington for two decades. He compromised with President Bill Clinton, and his instincts for realpolitik prompted him to side with the party establishment in one of the early face-offs with the tea party (in a special House election in New York). He may be a onetime rebellious backbencher, but when he became speaker of the House, his colleagues tried to push him from power.

The two men speak to contradictory desires within the party. The grass roots yearn for a fighter who is prepared to take on Obama in the most strident and confrontational way possible — to call him out as a socialist or worse. The applause Gingrich generates at debates with his defiant and indignant performances gives voice to this visceral urge among many conservatives.

But many in the party also know they need the steady competence of a leader who is capable of restraining the worst excesses of the hard-right activists and translating the conservative rhetoric and ideas that unite the party into a governing strategy that can bring the Republicans a White House victory in November and success beyond.

To some Republicans, those who choose not to run for president still seem more suited to the roles that Romney and Gingrich are now assigned. Former Mississippi governor Haley Barbour and Indiana Gov. Mitch Daniels would have traveled in the establishment lane, and yet both have the kind of conservative credentials and credibility that Romney lacks.

New Jersey Gov. Chris Christie can be as pugnacious as Gingrich and has been as rhetorically harsh about Obama’s leadership as any prominent Republican in the country. But he is not weighed down with the Gingrich liabilities.

Former Arkansas governor Mike Huckabee is another who could have played the role of populist conservative, as he did in 2008, when he showed he could espouse the most conservative of views but with the kind of humor and cheerfulness that softened the edges of his message.

No one can say that a Barbour or Daniels or Huckabee or Christie would have done better than either Romney or Gingrich had they chosen to run. But there is no mistaking the feeling that there is still a mismatch between the character and personality of the Republican Party today and the people who are seeking to lead it into the fall campaign.

There is no perfect answer either to the question of which of the presidential candidates is best suited to lead, although, ironically, Republicans this week can look to Florida for some hint of what the party really wants — an amalgam of former governor Jeb Bush and new Sen. Marco Rubio.

Bush, whose last name virtually disqualified him from seeking the presidency in 2012, might be running away with this nomination battle had he changed his name and entered. He has impeccable credentials as a conservative and served two terms as governor of this critically important swing state. He is sensitive to the GOP’s problems with Latino voters. He is respectful of the Reagan tradition but aware that the times demand something more than nostalgia for a bygone era.

Rubio is the young hope who so captivated the tea party movement as a candidate for the Senate that he drove former governor Charlie Crist not just to the sidelines but out of the party, and who in his first year in office has moved smartly with an eye toward a future that can bridge the gulf between the tea party and the establishment.

Neither has taken sides in the nomination battle, and when there were rumors late Saturday that Bush might be moving toward an endorsement, he quickly reiterated, in an interview with Bloomberg News’s Mark Silva, that he would remain neutral as the fiery contest between Gingrich and Romney moves into his state.

Barbour, Daniels, Huckabee, Christie, Bush and Rubio may spark the imaginations of hungry Republicans, but the ideal is always unattainable and politicians are never the same in practice as they are in the speculations of pundits or activists or the elites.

Today, the Romney-Gingrich contest has suddenly captured the attention of the nation, if only because it is so improbable. And the two are dividing up the party along predictable lines.

In South Carolina, Gingrich did better among evangelicals, among those making less than $100,000, among those who do not have college degrees, among those who see deficit reduction rather than job creation as the higher priority for the next president — a proxy, perhaps, for tea party enthusiasts.

Romney was stronger with those who have college degrees, who are not evangelical Christians, who prize job creation and who make more than $100,000 a year, although he has lost some of those groups to Gingrich.

Gingrich and Romney have attributes that have made them worthy of this showdown. Romney has, with some exceptions, been disciplined and patient. He acknowledged Sunday he had made a mistake in not releasing his tax returns more quickly in order to put the issue to rest, but in many other ways he has shown steadiness and intelligence and improvement as a candidate this time.

Gingrich has shown remarkable resilience, having blown up his campaign last spring only to rise again and then see his candidacy plummet under a barrage of negative ads and then, improbably, to prosper again. No one who watched him in Iowa anticipated he would win a double-digit victory in South Carolina.

But the other side of the balance sheet is what makes Republicans wonder what they might be getting, no matter what they get. Romney is gentlemanly and has not shown the rhetorical edge in attacking Obama that many conservatives seem to want. He has moved right in the nomination campaign but at heart is not an ideologically driven politician. Problem solving is his motivator.

Gingrich has plenty of over-the-top rhetoric. Two decades ago, he railed against the “corrupt liberal welfare state.” He has updated that to fit the times — hence his label for Obama as “the food stamp president.” But he is not a perfect tea party politician. He is not a pure small-government politician. His rivals say his economic and entitlement proposals would blow a big hole in the deficit. Some conservatives still won’t forgive him for attacking House Budget Committee Chairman Paul Ryan’s Medicare proposal.

Although they have a real opportunity to win the White House in November, Republicans may be a presidential cycle away from fielding a group of candidates who are more in sync with the real identity of the party, who represent the party’s future generation and who are more capable of bridging the cultural and stylistic gulfs that exist.

That is why there could be an inevitable letdown once this race is over and the party contemplates its leader for the fall campaign — or deep disaffection among the supporters of the loser. Finding the acceptable compromise has been frustratingly difficult for the Republicans. No one yet satisfies, but someone must. For Gingrich and Romney, this is the time to prove the doubters wrong.