Thus Mr Blair not only made her handbag famous, but, in the teeth of her growing reputation for bellicose divisiveness, put his finger on an aspect of values they shared in common.
Her handbag was back in Downing Street last Thursday, for a very different encounter. He has made it clear that her style of leadership is a model for him. She has basked in the glow of his admiration, returning the compliments with interest. This was the "most formidable" Labour leader since Gaitskell, a man who "won't let Britain down".
As a public relations coup, the meeting was sensational. It was the act of a supremely self-confident politician. It identified him with Britain's most forceful recent leader. If there was anything in her handbag this time, it was Essence of National Leadership, and she poured it into Mr Blair's tanks. She may never have been widely popular, but she was respected, and especially so by a large section of the electorate that Mr Blair won over on 1 May and on to which he intends to hold. The meeting sent an important signal to this group, all part of New Labour's professional after-sales service. And it sent a signal abroad: nice guy, can tell you how to win elections - but watch out. They all remember the handbag too.
There were other signals. Other former prime ministers have not been consulted. John Major's offer of advice on something he thought he was really good at - negotiating with European partners - has not been taken up. James Callaghan's telephone has been strangely silent, except when the BBC rings.
But what of the substance of the Blair-Thatcher summit? Is Mr Blair likely to have learnt anything? Sure, "she is still a figure on the world stage and knows many of the leading players", as a Labour source explained. But there is nothing she can tell him about how to handle world leaders that will not be in the Foreign Office brief. And, certainly, she has a reputation for resolution and firm decision-making, but her alleged advice to decide what is important and not fight on "ground on which you cannot win" is both banal and unnecessary. Mr Blair has demonstrated, as much in three weeks of government as in three years of opposition, that he is exceptionally sure-footed.
She did, however, have one important thing to say. That Robin Cook, she reportedly complained - he didn't even mention the Special Relationship with the United States in his "mission statement" for the Foreign Office (she cannot like mission statements, however much a product of her time they were).
It is true that the US did not feature in the five priorities of Labour foreign policy outlined by the Foreign Secretary. Although Mr Cook did promise, at his video-enhanced news conference, that relations with Washington would be better than under the Tories. And he had a sound basis for this claim, in the shape of Mr Blair's next important visitor to Number 10, Bill Clinton, who arrives on Thursday.
The conversation between Mr Blair and Mr Clinton is likely to be more substantive than last Thursday's courtesies. Whatever the cross-party affinities between Mr Blair and Lady Thatcher, Tony and Bill share a common "project" - coming from the left to govern from the centre. But she drew attention to one of many unresolved, subterranean questions about the new government: what has happened to Labour anti-Americanism? Mr Cook, one suspects, retains a stronger streak of it than some, whereas the New Labour core - Blair, Brown, Mandelson - are personally and politically pro-American.
This matters, because Lady Thatcher is right - although for the wrong reasons. There does need to be a strong British-American relationship. But what is special about it is that the United Kingdom both is in Europe and speaks English. Britain should have a pivotal role as the gateway between America and the European Union. It is not simply a matter of language, but that our two countries share fundamental beliefs about liberal economics. Whereas on the Continent liberty is a political concept, for Anglo-Saxon capitalists political freedom is founded on economic freedom. Lady Thatcher understands that, but cannot grasp that America is not interested in Britain except as a member of the EU.
That is a message Mr Clinton will underline this week. He and Mr Blair will discuss Northern Ireland and Hong Kong, but the real talking will be about Europe. For the President, a single European currency is a tricky local detail: the big picture is wider European integration, and he is in favour of it. He wants stability in central and eastern Europe, and Russia safely moored alongside. Last week he talked about Nato enlargement and the agreement with Russia giving the chance of "a framework of an undivided, democratic Europe" but added that this would need more economic - and, crucially, political - integration to succeed.
This is not what Lady Thatcher means by a Special Relationship, but, for all the presentational brilliance of last Thursday's "secret" chat, this Thursday's conversation in Downing Street will be a much better guide to the future.Reuse content