So what did President Bush make of it all? A presidential visit is a whirl of scattered impressions seen from the back of a limousine - and intense hours of conversation in formal rooms: a snatch of bunting on The Mall and planning sessions with Prime Minister Blair; an edited collation of evening news programmes and detailed briefings on the state of British public opinion.
From all that, he would take home hard facts and general impressions. He would have observed the continuing strength and intimacy of the Anglo-American alliance. He would have read the polls showing that a majority of British people support the war on terrorism - and his visit. He would have seen that important parts of the press vigorously support the war. He would have been forcefully reminded that Britain, too, is a terrorist target - and that the British are able to mourn their losses and to keep fighting.
Protesters in Sedgefield|
At the same time, President Bush would have perceived how very nervous his British hosts were about this visit. He would have understood why no public events had been scheduled for him - and why his car took him only as far as from the back door to the front door of Buckingham Palace. He would have apprehended the hostile political attitudes of the heir to the British Crown - and his acute political senses would have perceived the loosening of Tony Blair's grip on the Labour Party. Of course, he would already have known that many even in the Conservative Party oppose his policies and fantasise about outbidding Mr Blair for the anti-war vote.
It's an ambiguous picture, in other words, tinged with tendencies both good and bad from an American point of view. But there is at least one unambiguity that he would certainly have witnessed - and that millions of Americans will witness with him.
The culminating moment of the big anti-war demonstration on Thursday was the toppling of an effigy of President Bush, parodying the liberation of Baghdad on April 9. Demonstrators took turns stamping and trampling on the statue; as soon as they were done, they invited the MP George Galloway - of all the people in the world - to step to the microphones.
I spent three hours with the marchers on Thursday and talked to many dozens of them. Many ridiculed the "bubble" in which the president supposedly moved - and wondered, with varying degrees of sarcasm, whether anyone would tell him about their actions in Trafalgar Square. They seemed to take for granted that if Mr Bush - and Americans in general - saw what the demonstrators were doing, that Mr Bush (and Americans) would somehow be impressed. Talk about living in a bubble!
Europeans can become so fascinated with the subject of their feelings about America that they can sometimes lose sight of the - at least equally important - issue of American feelings about them. After a recent visit to America, the historian Timothy Garton Ash reported, "The current stereotype of Europeans is easily summarised. Europeans are wimps. They are weak, petulant, hypocritical, disunited, duplicitous, sometimes anti-Semitic, and often anti-American appeasers."
Thanks to Mr Blair's leadership and courage, Britain has been exempted from this indictment. But those people who parodied and scorned the triumph over Saddam's tyranny seemed all too eager to join the French and Germans in the dock.
What does Mr Bush make of it? The protests and jeers he drew on his visit will not cause him to rethink the rightness of his policies. He thought long and hard about them before he put them into effect, and having made up his mind, he will see them through - to the end. Instead, those protests and jeers will feed worries about the future direction of Britain.
The problem is not the old communists and ex-hippies who did the jeering. That old Left has been stranded by the tides of history - it is shrivelling and dying before our eyes. The problem is the new nationalisms now emerging in Europe that hope to exploit anti-American feeling for their own purposes: the new nationalism of the Eurocrats, who dream of building a superpower of their own; the new nationalism of the extremist fringe within Europe's Muslim minority, that makes common cause with antique peace groups in order to advance a quite different ideology and cause.
I asked everyone to whom I talked in Trafalgar Square to express for the record an unqualified condemnation of the murder of their fellow Britons in Istanbul. Not one of them could do it. "Of course, I condemn it," they would say, and then, with a pause that barely rated a comma, they quickly added some words to the effect of: "But you have to understand how we have driven the bombers to do such things."
The argument over President Bush's visit to Britain is not an argument about him at all. It is an argument about Britain and its future. The anti-Bush mood is a spur that some hope to use to hasten Britain ever further into the European Union and that others hope to use to legitimise their ill-concealed sympathies for the enemy in the war on terrorism.
The war in Iraq has become a war within the countries of Europe over their future identity - and that war is now being waged in Britain: the most stable, the most cohesive, the most unabashedly patriotic nation of the whole continent.
Which side will prevail in this war? The answer matters hugely to Americans. It matters even more to the people of Britain. And, if as President Bush wings his way homeward, he allows his always practical mind to drift toward abstract speculations, it is this question that will, and should, worry him most.
David Frum, a former special assistant and speechwriter to President Bush, is the author of The Right Man: The Surprise Presidency of George W Bush (Weidenfeld & Nicolson).