The move carries risks, say moderate Republicans. They cite polls showing substantial public support for a balance of tax increases and spending cuts to tame the deficit.
A survey by the centrist Republican Main Street group found that most Americans feel the Republican Party chiefly "looks out for rich guys," said the organization's president, Steve LaTourette. The former Ohio congressman said the party's self-examination, on balance, is healthy and essential, and he thinks Republicans outflanked Democrats in the sequester showdown.
Virginia-based Republican consultant Chris LaCivita said the intraparty debate is healthy and should reassure voters the GOP is vigorous and transparent.
Obama is appealing to Graham and other Republicans who say new taxes on the rich might be possible if Democrats agree to tackle the long-term funding problems of Medicare and Social Security. Most Republican leaders adamantly oppose any increases in income taxes.
Other internal debates that Republicans must resolve include gun control and immigration. Sen. Charles Grassley of Iowa was the only Republican on the Judiciary Committee this week to vote with Democrats to make it a felony to buy a gun for someone who could not pass a background check.
Regarding immigration, Romney and other GOP candidates did poorly among Hispanic voters, many of whom see the party's immigration policies as a slap at all Latinos.
Rubio, the son of Cuban immigrants, is heading a Republican effort to craft immigration legislation to counter Obama's proposals.
Another prominent Republican, former Florida Gov. Jeb Bush, created a stir this week by saying he did not support a pathway to citizenship for illegal immigrants, even if they have lived peacefully for years in the United States. Bush, whose father and brother were presidents, tried to soften the comments later, but they served to remind everyone that Republicans face difficult debates over immigration.
The 2016 presidential race could possibly attract Bush, Rubio, Ryan, Paul, Thune and numerous other Republicans, including Govs. Chris Christie of New Jersey and Bobby Jindal of Louisiana.
The 2012 Republican primary featured exhaustive and often unwieldy debates. Catering to hard-core conservatives who dominate primaries, Romney and his rivals veered to the right on many issues, a process that many feel hurt Romney among centrist voters in the general election against Obama.
With tea party-backed House Republicans thus far dominating the tax-and-spending debate, it's unclear whether the GOP can avoid similar clashes in 2016. Before it confronts that question, however, the party must move closer to consensus and intraparty peace over immigration, anti-terrorism and other matters.
This week's back and forth in the Senate indicates the soul-searching will go on a while longer.
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