Every political rebirth starts somewhere, and in the case of America's hapless Republicans, you could argue that somewhere was a modest pizza joint called Pie-tanza a couple of months ago. The occasion wasn't much – just three party luminaries at a restaurant in a strip mall in the Washington suburb of Arlington on a Saturday morning, taking questions from a few dozen voters. The occasion was billed as the first stage of a "listening tour" organised by a newfangled body of the National Council for a New America. More accurately, it might have been called "How to make the Republicans relevant again – any suggestions gratefully received."

In this summer of 2009, it is hard to exaggerate the Republican predicament. Out of fashion, out of power, and above all out of ideas, the party of Abraham Lincoln, Dwight Eisenhower and Ronald Reagan now counts as its most recognisable national figure a somewhat flaky, soon-to-be-ex-governor of Alaska named Sarah Palin, of few discernible political talents other than an inordinate ability to attract press attention. You may compare that predicament with that of the Tories after Labour had swept them from power in May 1997. Not only had the Republicans been in power too long; by last November's election, the country had grown as bored of them as Britain had of the Conservatives a dozen years ago. Both, moreover, had to cope with a charismatic new challenger: it may be hard to remember now, but Tony Blair once generated an excitement comparable to Barack Obama today.

Like the Tories, the Republicans portrayed themselves as defenders of old-fashioned, traditional America, only to suffer a string of sex scandals which exposed them to ridicule. Even now, they might have surrendered power, but not the capacity to embarrass themselves. In the past month alone, a couple of rumoured contenders for the 2012 nomination – John Ensign, a right-wing "family values" senator from Nevada, and Mark Sanford, the equally conservative governor of South Carolina – have been exposed as philanderers.

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Ensign, it emerged, had an affair last year with a campaign worker, then his parents made a "gift" of almost $100,000 to her family. But that transgression paled beside the Sanford melodrama. When the governor went AWOL last month, aides at first claimed he had gone on a solitary walking trip in the Appalachians. A few days later he re-appeared to admit at a tearful press conference that he had actually been in Buenos Aires for a tryst with a former Argentine TV journalist with whom he had fallen in love. For now, both men are clinging to their jobs, but any lingering claims by the Republican Party to be a beacon of moral rectitude are in ruins.

In some ways, the Republicans are in an even deeper hole than the Tories a dozen years ago. The debacle of John Major's "Back to Basics" campaign obscured the fact that he bequeathed to Labour a stable and expanding economy. ' Last autumn's financial crisis and the current recession in the US, by contrast, were mainly brought about by a Republican free-market philosophy run wild. The Crash of 2008 was a failure not just of economic management but of an anti-interventionist, "grab what you can" ideology that had dominated American politics for decades.

Even more than 1997 in Britain, 2008 in the US will surely go down as a watershed year, like 1932, when the harsh reality of the Great Depression and Franklin Roosevelt's New Deal ended early-century Republican dominance. The Democratic era that followed eventually disintegrated amid the turmoil of 1968 and a feeling that liberalism was running amok. But the subsequent conservative era in the US, with Ronald Reagan its patron saint, has ended in its turn – its fate sealed not just by the shortcomings of George W Bush but a deeper sense that it had nothing more to offer.

These eras were not monolithic: Dwight Eisenhower provided an eight-year Republican interlude in the heyday of the Democrats, while Democrats Jimmy Carter and Bill Clinton occupied the White House when the underlying tide was running in a conservative direction. Neither, however, reversed that tide. Carter was a one-term president elected mainly in reaction to Watergate, while Clinton was a master of compromise, or "triangulation" as it would be known, who did not shake the pillars of Reaganism. Indeed, a fair portion of the deregulation that brought about the 2008 meltdown took place on his watch.

But it is Republicans who are now paying the price. Ever fewer people identify with the party. A recent Gallup poll found that only 28 per cent of Americans considered themselves Republicans, compared with 36 per cent who described themselves as Democrats – the widest such gap in a quarter of a century. The independents who constitute the other third of the electorate lean in the Democrats' direction.

America's political geography has been transformed. Democrats today control not only the White House, but both chambers on Capitol Hill. In the Senate, Democrats have achieved a theoretically filibuster-proof majority of 60 for the first time in 30 years (Republican senators, in other words, will find it far harder to derail a bill with endless speeches from the floor). The country's electoral map has been similarly transformed. Democratic states have expanded from both coasts into the Midwest, and into the South. In the 2008 Presidential election, the states carried by John McCain were essentially a giant L, its base stretching across the deep South into Texas before turning north across the plains and Rocky Mountains to the Canadian border.

Ideologically, too, the Republican coalition has splintered. The party still does well in the sprawling suburbs. It remains the party of big business, identified with limited government and strong national security. But the so-called "Reagan Democrats", appalled by the anything-goes liberalism of the 1960s and 1970s, are now returning to their old fold.

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The attention lavished on social conservatives and social issues was increasingly at odds with the national mood. The endless "culture wars" over abortion, guns and gays mask the fact that Americans are pragmatists, not ideologues. Yes, the US is more conservative than Britain. But the Republicans moved too far to the right, and in the process lost touch with the real America, a country that is multi-ethnic, increasingly university-educated, and much more exercised by the everyday problems of jobs, healthcare and education than by the issue of whether gays should be allowed to marry. Democrats used to walk in dread of Karl Rove, George W Bush's top political strategist, who was boasting as recently as 2004 of a permanent Republican majority – even though Bush's narrow victory that year was above all a triumph of organisation. Five years on, the prospect is of a permanent Republican minority.

It is easy to mock that Saturday morning at Pie-tanza. The "listening" part was real enough, as a mostly Republican crowd vented its frustrations. But the answers from the three participants – the former Massachusetts governor Mitt Romney, the fast-rising Virginia congressman Eric Cantor, and Jeb Bush, younger brother of George W – were notably short on specifics, consisting mostly of slogans and platitudes you hear at a campaign debate.

And that was probably no coincidence. At least two of the three will surely figure in the contest for the Republican nomination in 2012. Romney, a sleek businessman who projects managerial competence, is already running in all but name, undeterred by his defeat by John McCain last year. Sharp-elbowed and clever, Cantor is only 46 but already the second ranking Republican in the House of Representatives. The Pie-tanza meeting was his idea. As for Jeb Bush, he was a highly successful governor of Florida. Affable and undogmatic, he remains very popular within the party. His problem, of course, is national Bush-fatigue. With any other surname, he might well already be the Republican frontrunner.

But if the road to redemption begins with the admission of sin, the trio at least made a start. As Bush put it, "The Democrats have something. I don't like it, but they have it." It was time, he said, for Republicans to stop trying to turn the clock back to the vanished golden age of Reagan, and to look to the future.

Alas, there are two conflicting visions of that future. As they survey the electoral rubble around them, the party faces an excruciating and critical choice. Does it become more moderate? Or does it move even further right – reasoning that the Republicans' mistake last year was not that they were too conservative, but that they were not conservative enough? No political species is more detested by conservative true believers than RINOs – "Republicans In Name Only" – whose lack of conviction is held to have played into Democrats' hands.

In fact, and for the reasons outlined above, Republicans have no choice but to move towards the centre, if they aspire to be a party of government rather than a minority cult. In a democracy – be it the US, Britain, France or anywhere else – elections are won and lost on the middle ground, populated by independents and other swing voters. Republican policies must appeal to these people, not just the loyal base. As Jeb Bush, whom no one would consider a RINO, said, "You can't beat something with nothing, and the other side has something."

But if the answer is obvious, the process of reaching it will be long, painful and divisive. One reason is institutional. Unlike Britain with its parliamentary system, America only has a true leader of the opposition for a few months in a presidential election year, between the emergence of a nominee and the election itself.

John McCain forfeited that role when he lost to Obama. The most senior members of the Republican hierarchy are Mitch McConnell and John Boehner, minority leaders of the Senate and House of Representatives respectively. But both are legislative tacticians, not electoral grand strategists. The party has no shortage of attractive figures: among them Bobby Jindal, the young governor of Louisiana and the first Indian-American to occupy a state house, and Jindal's peer in Minnesota, Tim Pawlenty. There's Newt Gingrich, architect of the sweeping Republican victory in the 1994 Congressional election, and never short of ideas, then there's Cantor.

And, of course, there's Sarah Palin. If social conservatives alone selected the nominee, Palin would win by acclamation. The star power she first displayed with her barnstorming speech as McCain's running mate to the 2008 convention is as strong today. The fevered speculation over her future, following her surprise resignation as Alaska's governor this month, merely confirmed that whatever else, she's news.

Whether Palin definitely plans to run in 2012, fancies making big money on the book and lecture circuit, or simply yearns for the quiet life, probably she herself does not know. But one thing may be stated with confidence: the Palin future, barring a quite inconceivable political makeover, does not include the Republican nomination for President, let alone the White House. Republicans may have lost their bearings, but not entirely their senses. Polarising, and with scant appeal beyond the right, Palin offers Republicans merely a one-way ticket to irrelevance.

Whatever she does will, moreover, not alter the fact that the party has no official leader, no David Cameron to take charge of the remaking of the party. Into this vacuum, others have stepped. For a while earlier this year, the loudest Republican voice in the land belonged to Rush Limbaugh, the sneering and bombastic talk-radio host. Then it was the turn of Dick Cheney, taciturn and secretive in his days as vice-president, but who for a few weeks this spring seemed to be on every cable channel in his new persona of battle-hardened elder statesman, excoriating Obama as a feckless novice whose squeamishness on torture and tolerance of rogue states were an invitation to terrorists to do their worst.

But Cheney is even more unpopular than his old boss (who, by the way, has moved into a new house in a smart neighbourhood of Dallas, where he's keeping a low profile and writing his memoirs). He is the Republican past, not the Republican future. In the absence of agreement on what that future can be, the party of opposition is doing what comes naturally – opposing Obama and everything he says and does.

Alas, this makeshift policy merely underscores the Republican dilemma. The refusal to compromise with Obama, the branding of the president as a dewy-eyed liberal, naturally delights the conservative-dominated base. But the polls – for the moment at least – suggest these spoiling tactics are exactly what voters feel is not needed when the country should be closing ranks to tackle the huge problems it faces. Every party, it is said, needs a spell in opposition, to find new energy and ideas. Barring an Obama mess-up of epic proportions in the next couple of years, or another 9/11 or equivalent foreign cataclysm, Republicans cannot realistically look to recapture the White House until 2016 at the earliest.

Already, though, the outlines of a new Republicanism are discernible. The most promising, and obvious, area is the economy. Americans are still enchanted with Obama the man, but less so with his economic policies. They dislike the bail-outs of the car industry and are deeply worried by colossal budget deficits stretching to 2020 and beyond. After the reckless tax-cutting and heavy spending and tax-cutting of the Bush era, Republicans have a golden opportunity to regain the mantle of fiscal rectitude and competent financial management.

On social issues, the party must become more inclusive. Many of its strategists already recognise that being a Republican and supporting one or all of gun control, gay civil unions, stem-cell research and a woman's right to choose are not mutually exclusive. Social conservatives will disapprove. There will be furious argument. Yet those same conservatives – and Christian conservatives in particular – are as alarmed as any liberal by global warming, climate change and the degradation of the environment. A less dogmatic Republican Party, more open to minorities and sensitive to their problems, will also be a greener Republican party.

Of course the party will not shed its trademark toughness on national security and defence. It will instinctively prefer the private to the public sector. It will be the party of individual rights and individual responsibilities. It will be healthily sceptical of state intervention, but will abandon the Reagan mantra that government, far from being the solution to America's problems, was the problem. In short, it will return to being what it used to be: a "big tent" party, competing with Democrats across the board.

To do this, of course, they will have to make some Democratic policies their own. But what is so wrong or unusual about that? After all Bill Clinton, the "new Democrat", stole Reaganite policies, most notably on welfare reform, to the dismay of his party's left. Tony Blair embraced elements of Thatcherism as he turned Labour into an electable party again. And now David Cameron is filching some of Labour's clothes as he hauls the Tories into the 21st century.

In the meantime, Republicans must be philosophical. Every political party outstays its welcome. Americans will tire of Obama-ism and the Democrats as they tired of liberalism after the 1960s, and have tired of Reaganism now. "This won't last for ever," Jeb Bush put it that day in Arlington. "I've seen conservatives move up and conservatives move down, liberals move up and liberals move down." Obama's astounding rise was "a tribute to our country" from which Republicans should take heart, he added. "That will happen to us, too."

With friends like these...

Rush Limbaugh

The radio talk-show host and de facto voice of grass-roots social conservatism has been a highly vocal critic of President Obama's attempts to reignite the US economy. In January, he said Obama "is talking about the absorption of as much of the private sector by the government as possible, from the banking business, to the mortgage industry, the automobile business, to healthcare. I do not want the government in charge of all of these things... I hope he fails." Republican National Committee chairman Michael Steele said Limbaugh's rhetoric was "incendiary" and "ugly" (though he later apologised to Limbaugh for his remarks).

Dick Cheney

In May, after President Obama reiterated his intention to close down the Guantanamo Bay detention centre, former vice-president Cheney tore into his speech, accusing Obama of compromising CIA interrogation techniques and claiming he was jeopardising American security. The outburst was widely criticised, not least by senior Republicans including former presidential nominee John McCain. "When you have a majority of Americans saying we shouldn't torture, I'm not sure it helps for the [former] vice-president to go out and continue to espouse that position," said McCain.

Pat Buchanan

A fortnight ago, the 70-year-old former senior adviser to President Reagan was invited on a current-affairs programme to explain his antipathy toward Obama's nomination for the Supreme Court, the Hispanic Sonia Sotomayor. Buchanan had criticised her nomination as an example of "affirmative action". He was asked to explain why 108 of the 110 Supreme Court justices have been white. "White men were 100 per cent of the people who wrote the constitution, 100 per cent of the people who signed the Declaration of Independence, probably close to 100 per cent of the people who died at Normandy," he said. "This has been a country basically built by white folks."

Audra Shay

Earlier this month, Shay, 38, the chairman of the Young Republican Party, responded to a Facebook posting in which an acquaintance, Eric Piker, referred to the need to "take this country back from all of these mad coons" with the comment, "You tell em Eric! Lol." Shay defended herself by claiming she was responding to an earlier posting on the page, and stated that she did not "condone racial slurs".

Sophie Vogel