* Republicans still seeking answers after Obama victory
* Party aims to appeal more to women and minorities
* Is it the conservative message or merely how it is cast?
* Republicans tout successes on state level
By David Lindsey and Samuel P. Jacobs
WASHINGTON, Feb 21 (Reuters) - If there is anything more tense than the relations between Republicans and Democrats in Washington, it might be what is happening between two prominent Republicans: strategist Karl Rove and former House of Representatives Speaker Newt Gingrich.
Three months after Republicans failed to oust Democratic President Barack Obama from the White House, Rove, Gingrich and other Republicans are locked in an increasingly bitter debate over how to revamp their party to appeal more to women and minorities.
The latest in a series of spats surfaced on Wednesday, as Gingrich gave a blistering critique of Rove's plan to create a fundraising group aimed at boosting moderate Republicans in congressional races - and blocking the rise of far-right conservative candidates.
"Handing millions (of dollars) to Washington-based consultants to destroy the candidates they dislike and nominate the candidates they do like is an invitation to cronyism, favoritism and corruption," Gingrich wrote in the conservative magazine Human Events. He also chided Rove - a former top aide to President George W. Bush - for his work in several losing races last fall.
Gingrich's shot at Rove reflected the tension that has been building among Republicans since the Nov. 6 elections, when Obama won with big margins of support from women and minority groups, including the fast-growing Hispanic population. It was the fifth time in the last six presidential elections that the Democratic candidate got the most votes.
Since then, Republican strategists say, their party has been wrestling with two questions: Has our message gotten too conservative for an increasingly diverse and moderate national electorate? Or, is it just the way we've been selling it?
The effort to answer those questions helps explain much of the party's political maneuvering since the election.
It's why conservative leaders who believe there is nothing wrong with the party's low-tax, limited-government message are looking beyond Washington's gridlock to emphasize places where tax-cutting Republicans are showing more promise: at the state level, namely the 24 states where Republicans control the governor's office and the legislature.
And it's why, even amid the partisanship in Washington, there have been gestures toward the political center that would have been difficult to imagine just a few months ago.
Four Republicans in the Senate, including Marco Rubio of Florida, joined four Democrats in proposing an overhaul of the nation's immigration laws. Some conservative Republicans, including House Majority Leader Eric Cantor, have indicated they would back a path to citizenship for children of illegal immigrants, an issue that for years has been a non-starter among conservatives.
Republican leaders in Congress have sought to tamp down internal rebellions over fiscal issues and harsh rhetoric on immigration and abortion that led some voters in November to see Republicans as obstructionists in Congress, anti-immigrant and anti-women. House Speaker John Boehner removed four fiscally conservative lawmakers from powerful committees in December.
And this month, Rove's American Crossroads "Super PAC" announced that it was forming the Conservative Victory Project to recruit more moderates to run for high office.
Rove acknowledged that the idea behind the new group was to avoid having candidates such as Representative Todd Akin of Missouri and Richard Mourdock of Indiana, Senate hopefuls last year who were backed by the conservative Tea Party movement.
During their campaigns, Akin and Mourdock, both of whom oppose abortion rights, made comments about rape that helped to sink their chances for election. Akin resisted calls to withdraw from his race after saying that victims of "legitimate rape" rarely get pregnant.
The comments by Akin and Mourdock were among several by conservative Republicans that often were cited by Democrats who accused Republicans of waging a "war on women."
"It's amazing that people think Todd Akin was the best we could've come up with," Rove said on Fox News. "We need to get better conservative candidates and win."
It is not unusual for a party to try to "re-brand" itself and its message after losing a presidential race.
But many staunch conservatives and Tea Party-backed Republicans are dismayed by efforts to reach out to Democrats or compromise on any front.
During the past two weeks, Rove and his new group have been a popular target for such Republicans.
Iowa Governor Terry Branstad called Rove's move a "mistake." Former Representative Joe Walsh of Illinois, a Tea Party favorite who lost his bid for re-election in November, said he plans to launch a conservative fund-raising group to battle Rove's group and "fight for the soul of the Republican Party."
And Akin, defending his campaign from Rove's criticisms, accused the strategist of trying to "get rid of conservatives."
Meanwhile, a Tea Party group that had sent out a fund-raising e-mail with an altered photo showing Rove in a Nazi military uniform apologized to him on Tuesday, saying that a vendor had doctored the photo without the group's approval.
Rove said he does not want to fight with the Tea Party and other conservative groups, and that he is simply trying to put Republicans in position to win elections.
That sentiment was echoed by Republican strategist Ron Bonjean.
"With a void of power after the election, the struggle between pragmatists versus purists on electing Republican candidates seems inevitable," Bonjean said. "However, it's not about being conservative or moderate. The best conservative with the best chance of winning the election should have the backing from everyone."
WATCH THE GOVERNORS
In recent weeks, Republicans have made a point of touting their efforts on the state level as models for how the party can improve its appeal to voters on a national level.
Republicans now hold the governor's office in 30 of the 50 states. In 24 of those states, the legislature also is controlled by Republicans.
Party strategists say that anyone looking for clues about what they will emphasize in coming elections should examine states such as Michigan and Louisiana. If they are a guide, Republicans' long-term strategy does not include backing away from conservative platforms.
In Michigan, Governor Rick Snyder champions "right to work" laws that limit unions' rights, measures that Republicans say boost economic development.
Louisiana Governor Bobby Jindal is seeking to eliminate the state's personal and business income taxes, and replace them with more sales taxes.
Jindal says the plan would expand the state's tax base, create more revenue and boost the economy.
In Indiana, former congressman and new governor Mike Pence has made cutting the state's income tax by 10 percent a legislative priority. And in Kansas, Governor Sam Brownback is pushing to eliminate the state's income tax. Both support raising sales taxes.
Democrats and other critics say the emphasis on sales taxes would shift more of the tax burden to lower-income people while reducing taxes on the wealthy.
But Republicans counter that tax-cutting measures will have broad appeal - and ultimately help Republicans win more support among women and minorities - because they will stimulate the economy and create jobs.
"It's the governors who are coming up with good, creative ideas," said New Hampshire Republican strategist Dave Carney, who advised Texas Governor Rick Perry's presidential campaign.
Joe Allbaugh, who managed George W. Bush's presidential campaign in 2000, agreed.
"I don't see anybody in Washington who is going to catch fire" in the next presidential race, Allbaugh said. "It's very likely that the positions taken by a handful of Republican governors will determine the shape of the party's ... positions for 2016." (Editing by Marilyn W. Thompson and Lisa Shumaker)