Over the years the symbols and settings have changed - from tubes of Colgate toothpaste to an offhanded "Yo, Blair" on an unsuspected open microphone at an international summit. Back in February 2001, amid the wood cabins of the US President's retreat of Camp David, our Prime Minister was wearing "ball-crushingly tight blue corduroys", as the British ambassador fetchingly put it. His host had donned a typically macho leather pilot's jacket.
Last week, the props were very different in the gilt and stucco splendour of a tsar's palace in Russia: suits, ties and a formal lunch to round off the latest gathering of the G8 powers. But after five and a half years in which the alliance between the two men and their two countries has helped to change the world, the same mystery persists. What, precisely, are the ties that bind the unlikely couple of Tony Blair and George Bush? Is it a shared world view, a common sense of mission, or realpolitik and self interest? Or could it even be God?
I must confess a tiny personal connection to this fascinating tale. It was my wife, a correspondent for Reuters, who asked the obvious question at that first meeting in 2001: what did these two men, apparently so different, have in common? A pause, then Bush replied, "Colgate toothpaste". Careful, Blair shot back, "they're going to wonder how you knew that". A frisson, as they say, ran around the assembled British press corps. Did the British Prime Minister and the American President share a bathroom? The immediate mystery was quickly solved. Tubes of that particular brand of toothpaste were distributed to every residential cabin at Camp David. But five and a half years on, the question remains as tantalising as ever.
The "special relationship" has seen some odd pairings. The imperialist, conservative Churchill started things, with his extraordinary friendship with Franklin Roosevelt, a Democrat determined to ensure the break-up of the old European colonial empires. A couple of decades later, Harold Macmillan contentedly played wise uncle to John Kennedy, a generation his junior and again from the opposite side of the political fence.
Later personal alliances made greater ideological sense. Ronald Reagan and Margaret Thatcher were cut from similar cloth, as were Blair and Bill Clinton. But Blair and Bush? How did the co-author of the third way, the Oxford-educated exponent of a modernised European social democracy, become partner and confidant of a rough-hewn, resolutely anti-intellectual Texan?
One might also wonder how this embodiment of the "special relationship" squares with the languid musings of Macmillan to Richard Crossman (at least as recounted by the latter), as America entered the Second World War. His words still contain more than a grain of truth about British attitudes to the US. "We, Crossman, are the Greeks in the American Empire. You will find the Americans much as the Greeks found the Romans - great big bustling people, more vigorous than we are."
Blair has never indulged in such patronising melancholy. Indeed, every sign is that he fancies himself as a bit of a Roman too, as the leader of an old and wise country, to be sure, but also one that still has a bit of bustle, and which has not lost the will to act.
Nor is the Prime Minister's seamless transfer of affection from Clinton to Bush as strange as all that. For one thing, Clinton himself is said to have advised Blair to do precisely that. "Be his friend. Be his best friend. Be the guy he turns to." For another, the Clinton-Blair partnership was not perfect. Ideologically, they may have been soulmates, says Blair's biographer John Rentoul. "But like many other people, Blair found Clinton impossible to pin down. Bush, on the other hand, is a plain dealer, you know where you are with him."
Indeed, the whole "Yo, Blair" dialogue only serves to prove again that, with this President, what you see is what you get. This Bush behaves in private as he does at press conferences. He is impatient with details, he relies on intuition rather than a laborious sifting of facts to reach his judgements, and he expects to get his way. Clinton's world was painted a myriad shades of grey. Bush's (and to a lesser extent Blair's) universe is coloured black and white.
More important, their moral take on world affairs is remarkably similar. Moral, I repeat, not religious. Both are practising Christians. Both share an unshakeable conviction that they are right. That belief shone through their last joint press conference at the White House in May, even as Iraq was sliding into undeclared civil war. The pair were asked what mistakes they had made. Bush grudgingly admitted that his penchant for Texas frontier talk ("bring 'em on" and such) hadn't helped. Blair predictably went deeper, suggesting the American-British alliance had mishandled the issue of de-Baathification. But his faith was intact. The real problem, he said, was the blind ferocity of the insurgents in their opposition to democracy and freedom. The basic enterprise was no less right in May 2006 than in March 2003.
Nor was the Prime Minister a sudden convert to the merits of force. Back in April 1999 - when Bush wasn't even an announced candidate for the White House - Blair was arguing in a speech in Chicago that in an ever more interdependent world, the international community sometimes had the right to intervene in the affairs of another country. Nato had then just launched its bombing campaign in the Kosovo war, and the Serbian dictator Slobodan Milosevic was one of two individuals, both "dangerous and ruthless men", whom Blair singled out by name. The other was Saddam Hussein. Almost four years before the invasion that will for ever bind his place in history to that of his friend George, the Prime Minister's sights were set on Iraq.
Last but not least is Blair's belief in the absolute primacy for Britain of the transatlantic relationship, more important even than ties with Europe. He set out this thinking in a little-noticed but seminal address in January 2003 to British ambassadors assembled at the Foreign Office. It is worth quoting verbatim.
What, he asked rhetorically, "are the principles of foreign policy that should guide us? First we should remain the closest ally of the US, and as allies influence them to continue broadening their agenda." But, he went on, "the price of influence is that we do not leave the US to face the tricky issues alone". Foremost among these "tricky issues" was Iraq and its then presumed stockpiles of weapons of mass destruction. "Unless the world takes a stand on this issue ... we will rue the consequences of our weakness. America should not forced to take on this issue alone."
If so, the US might subsequently be tempted to yield to the other half of its nature - and turn away from involvement in the world. As Blair put it a few months later to a group of US correspondents in London, "the thing I fear is not American unilateralism, it is actually American isolationism, were it ever to go down that path". In short, hug them close - be they philandering third-way centrists or born-again Christians who "don't do nuance".
Such is the grand bargain at the heart of Britain's understanding of the "special relationship". The question is, has Blair hugged Bush too closely and received too little in return? He would argue he has differed with the US on Kyoto and global warming, on how to tackle world poverty, and that public loyalty has earned private influence. But the supporting evidence is pretty thin.
Blair did win agreement from Bush in early 2003 to seek a second UN resolution specifically authorising military action against Iraq, to give him desperately needed political cover at home. Much good that did. Nor did Blair's pleadings for greater US involvement in the Middle East crisis get very far - look no further than the current sorry mess. One way and another, our Prime Minister has gained little in return for his agreement to join the invasion of Iraq - without which, some argue, Bush might never have gone to war at all.
True, his popularity in the US has soared even as it has plunged back at home. Despite the turmoil in Iraq, 66 per cent of Americans say they have confidence in Blair as a world leader, and only 50 per cent in Bush, according to a poll last month. However, even the most heartfelt gratitude is small reward on its own.
And neither gratitude, nor the similar world views of the two men, nor even Blair's belief that the US relationship transcends all else in Britain's foreign policy, can quite explain why he has been ready to take so much grief over his unswerving support for the most unpopular American President of modern times - from the poodle jibes and tabloid headlines of "Walkies" deriding his visits to Washington, to the deep hostility within sections of his own party and a domestic approval rating that makes Bush's look healthy.
The answer may be very simple. Could it be that they actually get on well? Blair likes the President's directness - the knowledge that once his mind is made up, it will stay that way. For his part, Bush took to the Prime Minister's charms from the outset, and his appreciation has only grown since. Blair leapt to America's side after 11 September 2001 and flew to Washington days later to express Britain's solidarity. On Iraq he has never wavered. There is nothing that the Bush clan values as much as loyalty.
Over the past five and a half years, the two have met dozens of times, and will do so again on Friday in Washington. They have exchanged letters, and probably have made dozens of video-conferenced phone calls between the White House and Downing Street. On occasion there is a "good cop, bad cop" routine. At a press conference Blair will be the smooth-tongued voice of reason. Bush, meanwhile, plays the heavy - short on words, but long on the Marines.
There was a bit of that during their joint appearance in May. But that evening in the White House, Blair seemed oddly subdued, perhaps because - as one British journalist reminded him with a cruelty an American counterpart would never apply to the President - this might be his last trip to Washington as Prime Minister. He was submissive, visibly the junior partner.
The pictures and unguarded words during the microphone incident in St Petersburg told a similar story. Bush is chomping on a bread roll before lunch, having just complained about having to listen to boring, overlong speeches, and expressing surprise that Hu Jintao, the Chinese leader, also had an eight-hour flight home. Blair is hovering behind him, like an anxious retainer, his presence acknowledged by a breezy "Yo, Blair". Then, after some banter over a pullover that Blair gave the President (presumably for his 60th birthday earlier this month), they get down to the business of the Middle East.
Blair offers to go on a personal mission "to see what the lie of the land is". But Bush quickly quashes any notion of such freelance diplomacy. "Condi" (Condoleezza Rice) would go herself when she sees fit (in fact, she sets out today). Fine, Blair meekly replies, he was only trying to help. Just who is boss in this particular incarnation of the "special relationship" is as starkly revealed as in the imbalance of forces in the field in Iraq.
So, of course, it has always been. Reagan and Thatcher too might have seemed relatively equal partners - but then they did not fight a war together. Indeed, one mini-war divided them IN 1983, when the Americans invaded the former British colony of Grenada, a member of the Commonwealth, to topple a left-wing regime to which Washington objected. The Iron Lady was furious, but there was nothing she could do.
This more recent invasion was a joint effort, however lopsided. It has turned into a disaster - some would say the biggest blunder of American (and by extension British) foreign policy in half a century. But on Friday, even as Blair urges the dispatch of a robust international force to Lebanon, he will be defending the Iraq adventure. Truly, from tiny tubes of toothpaste not just beautiful, but also earth-shaking friendships grow.
POLICY SHIFTS: What Tony Blair wants - but hasn't yet got
Tony Blair's aides like to say he doesn't arrive at meetings with George Bush "with a shopping list".
The truth is that the British government is always seeking to gain concessions from Washington. But despite providing staunch support to the US on almost every international issue, it has gained precious little in return. To rebalance the "special relationship", the US would have to give way on the five following areas.
WEAPONS TECHNOLOGY: Despite the fact that US and UK armies fight shoulder to shoulder across the world, America jealously guards its military advances. Promises to grant Britain an exemption from stringent US arms-export controls have been broken time and again.
IRAQ: Britain has repeatedly urged the US to pursue less draconian detention and policing policies, and to restart stalled reconstruction policies, but to little effect.
EXTRADITION: New arrangements that allow UK citizens to be tried in the US for crimes committed on British soil are not yet mirrored by reciprocal arrangements.
WORLD TRADE: US protectionism - especially in agriculture - is one of the main stumbling blocks that has frustrated Tony Blair's hopes of delivering a massive boost to Africa's economy.
CLIMATE CHANGE: Although George Bush has moved a little towards an acceptance of the reality of climate change, there is no sign that the US will match UK commitments on reducing carbon emissions.
'IOS' POLL: The poodle question
Most British voters regard the Prime Minister as the "poodle" of the US President, according to an exclusive poll for the 'IoS'. And half of those interviewed say the closeness of the Blair-Bush relationship has damaged British foreign policy in the Middle East.
It is fair to describe Tony Blair as George Bush's poodle?
Agree 54 per cent
Disagree 37 per cent
Blair's close relationship with Bush stops Britain playing a more constructive role in the crisis between Israel and its neighbours.
Agree 50 per cent
Disagree 33 per cent
CommunicateResearch interviewed 1,010 adults by telephone last weekReuse content