Compassionate conservative of the centre, or cleverly camouflaged right-wing ideologue? A vocal champion of a new political bipartisanship, or a hardline wolf in sheep's clothing? Republicans would like him to be seen as the party's centrist, pragmatic answer to Clintonism.

But the truth is, the spin and the substance simply do not agree. Today's inauguration sees the enthroning of a Bush-Cheney team described separately by conservative columnists George Will and Lawrence Kadlow as the most conservative since Calvin Coolidge.

And a bitter Senate confirmation process over two nominations, for Attorney General and Interior Secretary, suggests a yawning gulf between the administration and its own party's moderates. On Thursday the liberal Republican Senator Arlen Specter was reduced to begging witnesses from the women's and abortion movements to get involved in the Republican Party to help move its centre of gravity.

"It's conservative to cut taxes," Mr Bush has argued. "It's compassionate to let people keep more of their own money." What George Bush has done, unlike Clinton, is not to drag his party to the centre ground of politics, but successfully to repackage an essentially unchanged party and message for the centre.

Mr Bush, however, is no ideologue (no intellectual either - he still only reads executive summaries of documents). His instincts are all to the right but at heart, like his father, one gets the sense he is driven more by a desire to rule or to manage than to achieve a particular end, to travel rather than arrive. The Bushes are said to be more loyal to people than ideas.

His propensity to mangle the English language is legendary, but what worries commentators is that it's often not clear if this is a form of verbal dyslexia or reflects chasms of ignorance. To say "We cannot let terriers and rogue nations hold this country hostile"; or of a death penalty case: "This case has had full analysation . . . I understand the emotionality of death penalty cases," is almost endearing and harmless.

But it's not reassuring to have an economic supremo who confuses trillions of dollars with billions and who appears to see key concepts in almost childlike terms: "A surplus means there is money left over. Otherwise it wouldn't be called a surplus"; and "It's clearly a budget. It's got lots of numbers in it."

To be fair, Mr Bush has a nice line in self-deprecating humour; most of it, however, funny precisely because of its truth. James Bennet writes in a profile in the New York Times that those who have not studied him closely "tend to divide into two camps, the inspired and the appalled". Yet it would be wrong to underestimate both Mr Bush and his talented administration.

If he lacks depth in the policy area he is nevertheless a hugely successful politician with empathy - many say an instinctive leader - who appears to understand his own limitations and surrounds himself with those who can answer his many questions. He prides himself as a decisive problem-solver who leaves the detail of implementation to others in a way that Clinton found impossible. "I'm no finetuner," Mr Bush once said.

The first US President with an MBA, although acquired in those distant days of his dissolute youth, he told journalists during the transition that "a good executive is one that understands how to recruit people and how to delegate". He expects loyalty and silence from his staffers, but when things go wrong is not one to cast around for someone else to blame.

The cabinet is crafted to fill the gap left by his weaknesses, and it is clear that VicePresident Dick Cheney will play a more significant role than any of his predecessors, including Al Gore.

Mr Bush's appointments, particularly of former army general Colin Powell to Secretary of State (effectively foreign minister), Dick Rumsfeld to Defence, and Paul O'Neill to Treasury, have done much to reassure the political and media establishment that the ship of state is in safe hands. But they may also push Mr Bush towards a more pragmatic politics.

In Texas he was famous for staying focused on the small handful of priorities on which he campaigned, and he is likely to do the same as President: tax cuts, education, reform of Medicare, and bolstering the military, particularly the controversial National Missile Defence.

The tax issue is where he is likely to clash with Congress first, testing his commitment to bipartisanship. Although Democrats now admit the soaring surplus leaves a margin for tax cuts as well as their preference for repayment of the national debt, there is still a gulf between them and the Bush target of $1.3 trillion in tax cuts over nine years.

Mr Bush sees the cuts as central to his entire economic strategy, a sort of magic pill whose heavy dosage is necessary to curb the recessionary tendencies in the economy, and he insists there is no room for compromise.

But many economists here read the recent Federal Reserve rate cut as a signal from its chairman, Alan Greenspan, that Mr Bush should perhaps ease off on a matter very dear to the Republican right.

The Treasury Secretary, Paul O'Neill, a businessman with a reputation for independence of mind, may well also undermine the incoming President's hand. Mr O'Neill at his Senate confirmation hearings this week played down the role of tax cuts "as a major component to drive the economy". He said they could be an "added tool" but interest rate movements by the Fed were a better tool for managing the economy.

In education, a field Mr Bush genuinely knows about and in which he can boast some success, his appointment of a Texas school superintendent, Rod Paige, has been widely applauded across the political floor.

With the word "vouchers" no longer in the political vocabulary, there does appear to be much room for common ground in improving standards through more rigorous testing and improving choice. (It appears even Mr Bush's campaign team was infected by Bushisms - his campaign website listed as "No 3 priority - Putting Education First").

One of the most bitterly-fought battlegrounds for the new administration will be the environment, particularly the protection of the Alaskan wildlife refuge, where Mr Bush and his controversial Interior Secretary, steward of the great US wildernesses, Gale Norton, believe oil exploration should be allowed as a priority.

Mr Bush, who, like his father and his vice-president, is steeped in the oil business, insists the only way to deal with the world energy crisis is for the US to become fuel self-sufficient, and that means reversing a 15-year protection of the Arctic. "There's going to be some property in these giant chunks of land that we can use and not damage the environment," he maintains.

Ms Norton is deeply feared by the environmental lobby. She is no passive administrator but an activist who comes from the deregulation school of her mentor, the Reagan Interior Secretary, James Watt, a classic hate figure to greens. She has been central to the creation of a vigorous conservative environment lobby which has sought to counter traditional regulation with market-driven incentives.

A former Colorado attorney general, she claims to be an avid walker and skier with a fondness for the "beautiful and special places", but she is impatient with state curbs on the rights of property-owners and is even being opposed by the mainstream Republican environmentalists.

In 1991 she even spoke of "a right to pollute"; not a carte blanche for polluters, but a right to be compensated should state environmental regulations interfere with the free use of property. But if the state has to compensate it is going to be far less keen to legislate.

Both she and the Bush nominee to head the Environmental Protection Agency, Governor Christie Todd Whitman, are committed to a root-and-branch review of the string of new regulations and orders issued by President Clinton in the last few months. These range from the protection of new swathes of national parks to clean-air rules for big trucks and buses which, they argue, add to the costs of industry.

Ms Whitman also made it clear at her Senate confirmation hearing that she intends to give states more leeway in implementing federal rules - code, in effect, for relaxing them.

The theme of states' rights is an important current in this transition process. "Texans can run Texas," Mr Bush has been wont to say, and he instinctively trusts governors and state legislators more than Washington politicians.

But the sub-text is also that this is one of the key battlegrounds between left and right. The former has since FDR used federal legislation to leverage up basic rights across the whole of the US, and undoing that work is easier at state level.

Appointments to the courts recommended by the hugely controversial Attorney General, John Ashcroft (if, as appears likely, he is approved), will face rigorous Senate questioning on this as well as on the big constitutional questions of abortion, equality and gun control. A long running war of attrition between the executive and the legislature is to be expected.

Mr Ashcroft's nomination has become the real lightning rod of Democratic opposition, partly because the early departure of Ms Linda Chavez deprived them of an alternative target, but partly because the man's own record has raised fears even in Republican ranks of how he will use his discretionary powers. His nomination is very revealing of Mr Bush's perception of his natural political allies.

In foreign policy the caution and pragmatism of Gen Powell, a figure whose political popularity gives him a public authority on a par with the President's, make any sudden rift with traditional allies unlikely. Lengthy discussions are promised before any troops are pulled out of Kosovo and before the National Missile Defence (NMD) system is deployed.

But the Republicans perhaps underestimate the opposition to NMD of both the European allies on the one hand, and Russia and China on the other.