We will have to wait for his memoirs before we know what impression Britain left on Barack Obama – and even when he gets round to writing them I suspect that this exceptionally well-mannered man will be too polite to be entirely frank. But the American president has certainly made an impression on Britain. The state visit has left us a country divided: split between the gushers and the gnashers.
For the gushers, the Atlantic alliance has been forged anew in the glowing crucible of royal ceremonial and barbecued sausages. The gushing was led by President Obama himself and David Cameron who jointly endeavoured to dispel cynicism about the "special relationship" by telling us that it is "stronger than it has ever been".
To the gnashers, all the platitudes, the pomp and the ping pong were cheap – or, in the case of the Buck House banquet, rather expensive – confections of public relations which demeaned the host country. The fawning over the American visitor by the British political elite has stirred the stomach acids of some observers from both left and right. One Tory commentator goes so far as to call it "a national embarrassment".
I am generally a gusher about Obama as a politician. I reckon America is fortunate to have a thoughtful, pragmatic, reforming leader who can make music with words. I think the world is broadly lucky that America is led by such a leader at this time. Even so, I can nod along with some of the response of the gnashers to his address in Westminster Hall. The speech was a decent one. Not the greatest piece of oratory he has ever delivered, but Mr Obama on a sub-optimal day is 10 times better than most politicians on their very best day. He produced a line that shone: "We believe not simply in the rights of nations, but the rights of citizens." He asked an interesting question. If America, Britain and their European allies did not champion freedom: "Who would take our place?" And he answered it with the bold – but essentially correct – assertion that, despite the rise of China, India, Brazil and the other emergent powers, the western alliance will "remain indispensable to the goal of a century that is more peaceful, more prosperous and more just".
When he'd received his standing ovation, it was mildly cringe-inducing to witness gnarled British politicians jostling like lovesick teenagers to grab the hand, exchange a few words or just touch the hem of the one. At the same time, it was easily understandable. As a witty man once said: "Politics is showbusiness for ugly people." So when average examples of the species are in the presence of an alpha politician who is handsome, charming and eloquent, who has an attractive and inspirational wife, and is furthermore in charge of the richest nation on the face of the earth, these lesser members of the breed are going to fawn like mad.
For sure, there was choreography down to the last flipping burger in the garden of Number 10. Craig Oliver, the prime minister's director of communications, and his American counterparts earned their pay cheques last week. You will be seeing the pictures of Obama and Cameron high-fiving again. And again. And again. David Cameron wouldn't be human – he certainly wouldn't be a politician – if he didn't enjoy glowing in the reflected sunlight of a politician who is more popular and powerful than any of the homegrown variety.
For the man from the White House, limbering up for next year's contest to stay there for a second term, London was one leg of a pre-election tour. He had come from sipping Guinness in Ireland and rediscovering the Moneygall roots of the O'Bamas. He travelled on to Warsaw. Psephological detail: there are 10 million Polish Americans. If my crystal ball is not lying to me, he is already pretty assured of re-election. And I thought that before the killing of Bin Laden made it impossible for Republicans to sustain their charge that he is too professorial and prevaricating to be commander-in-chief.
He is boosted by coverage on the US networks looking relaxed and authoritative on the world stage. It does no harm being pictured with America's favourite British couple – not Dave and Nick, but the Duke and Duchess of Cambridge. None of his Republican rivals will be able to say that they have addressed Parliament or bed and breakfasted at Buckingham Palace. It is worth noting this electioneering dimension, but it is just silly to get upset about it. Complaining that politicians are interested in staying in power is pointless.
If we cut through the cynicism of the gnashers and the sycophancy of the gushers, what did we learn about Britain and America? The barbecue bonhomie could not entirely mask the underlying tensions, especially over the conduct of the conflicts in Afghanistan and Libya. We could also see, at their occasionally awkward joint news conference, that president and prime minister are not soul brothers. How could they be? One is a Tory from leafy Oxfordshire; the other is a Democrat from Chicago by way of Hawaii.
Obama ignited the most spontaneous applause from the crowd in Westminster Hall when he hailed both countries as places where citizenship was not based on ethnicity, but on a set of democratic ideals: "It is possible for the sons and daughters of former colonies to sit here as members of this great parliament and for the grandson of a Kenyan who served as a cook in the British army to stand before you as president of the United States."
By contrast, David Cameron is the grandson of an English baronet who was descended from William IV. The president let slip an interesting statistic when he told us that, since David Cameron became prime minister, they have met or spoken on the phone "two dozen times". That is roughly once a fortnight. Not cold, but not that sizzling hot either.
Yet despite personal differences between the two men, they have quite a lot of points in common. The first priority for both is to sort out their domestic economies. Neither much cares for isms. Both are fairly easy people to deal with. Given the nuttiness of so much of the right in America, Barack Obama must wish more of his conservatives were like David Cameron.
Both came to power as avowed sceptics about military intervention. Mr Obama had called the Iraq war "dumb". Mr Cameron's most memorable sound-bite about foreign policy was: "You can't drop democracy from 10,000 feet." Office and events have reshaped both men and left them in a broadly similar place. Both are pretty desperate for a exit from the war in Afghanistan. Both have staked a lot on the Arab Spring. Neither will be able to claim a success in Libya until they have achieved the removal of Muammar Gaddafi. Both were agreeably surprised when, at the subsequent G8 summit, Russia changed position and declared that the Libyan tyrant has to go.
The phrase "special relationship" has become a cliche. Yet like most cliches, it contains a truth. The bonds between America and Britain are deserving of the adjective special. This goes beyond a common language, overlapping cultures, shared myths and an entwined history. After the obligatory jokes about the foolish war of 1812, during which we burnt down the White House, Obama observed that this is "one of the oldest and strongest alliances the world has ever known". He's not wrong. For Washington, Britain remains the most dependable of partners: the friend that instinctively reacts in a similar way to international crises and the first to call when the US is seeking to assemble a multi-national coalition. Of all the current tensions in the relationship, the biggest springs from American anxiety about the scale of the cuts to the British defence budget. For London, America remains the friend to turn to when serious things need doing. It was with Nicolas Sarkozy that David Cameron pioneered the idea of imposing a no-fly zone to prevent a massacre in Benghazi. But they had to have America with them to make it happen and have not concealed their frustration that the United States won't put more assets into that action.
The history of this relationship has often been told in terms of the dynamics between the people at the top: Churchill & FDR, Jack & Mac, Ronnie & Maggie, Bill & Tony and, less happily, Tony & George. But this alliance has not endured for so long and through so many ups and downs because individual prime ministers and presidents have happened to hit it off or shared ideological agendas. It is because Britain and America have had and continue to have a coincidence of interests.
More than that, Obama spoke another truth when he rooted it in a shared commitment, imperfectly practised but sustained for many years, to liberty and democracy. At the news conference, he put it quite baldly: "The United States and the United Kingdom have a unique relationship and that is going to be consistent regardless of who the president and the prime minister is." He was telling David Cameron, standing by his side as this was said, that the British prime minister could fall out of a helicopter and it wouldn't make any substantial difference. Perhaps that wasn't terribly diplomatic. But it was right.