The event took place behind closed doors just weeks before the opening of a party conference season which has seen Liberal Democrat and Labour politicians playing to their respective galleries. But it presents a more accurate picture of the state of relations between the two parties than last week's semi-orchestrated, low-level bickering.
On Tuesday, on the stage of Brighton's large, anonymous conference centre, Mr Blair - the first Prime Minister to address a Labour conference for 19 years - will use his unchallenged authority to underline the need for more modernisation of the body politic. The rhetoric will be uncompromising. If he sticks to early drafts of the speech, the Prime Minister will argue that this is "an age of extraordinary change and challenge", and that there is a "quiet revolution taking place led by the real modernisers - the British people".
In private his closest aides are, if anything, more enthusiastic about links with the Liberal Democrats. "No one knows where this will end," says one. That leaves open the possibility of full coalition, of a government where, as in Germany, senior ministers are drawn from different parties.
Sixteen years after the SDP famously promised to break the mould of politics, a less dramatic, and rather more piecemeal and typically British process is taking place. The spur has not been the creation of a new party, but the raft of constitutional changes introduced by an old one. In Westminster, local government, the regions and Europe, Britain's political parties are barely waking up to the forces Mr Blair is unleashing. The old party structures are certain to come under strain as, at more and more levels of politics, the age of coalitions beckons.
The process began almost immediately after Mr Blair's election to the party leadership in 1994. Under his predecessor, John Smith, senior Labour figures like Jack Straw would regularly attack the Liberal Democrats. Overnight that changed. The Blair logic was compelling even to sceptics: if Labour was trying to lure voters in the south, was it really sensible to alienate those who had been voting Liberal Democrat?
At Westminster, contact between Mr Blair's office and Mr Ashdown's took place before each Prime Minister's Question Time. Lines of attack on John Major were co-ordinated. Even during the General Election the channels remained open. Officially the Liberal Democrats will admit to contacts over the campaign in Tatton, and over the unsuccessful efforts to set up a TV debate between the party leaders. But other senior Liberal Democrats concede that there was pretty regular intelligence about what "our friends" were up to.
Then, after the General Election, came the now-publicised meetings between the two party leaders, and the invitation to the Lib-Dems to join a Cabinet committee on constitutional reform. In addition, Archie Kirkwood, a Liberal Democrat MP, became the chair of the social security select committee (the first such appointment of recent years).
Mr Blair's close friends insist that there could well be more to come, with Liberal Democrats being brought in as other issues arise. There is no barrier, it is implied, to Liberal Democrats joining Mr Blair's government as ministers.
This, so the argument goes, depends on how the Liberal Democrats play their hand, whether they are willing to abandon "knee-jerk oppositionism'' - calling, for example, for more money for health and education - in favour of playing a fuller role. At this point the minuet being played by the two party leaders becomes particularly tricky. For Mr Ashdown, with his own party to lead, and power bases to satisfy, dropping a flagship policy will not be easy. For his part, Mr Blair may not be able to tolerate too vocal an opposition from those he has gone out of his way to bring on board. There is, then, potential for things to go wrong.
Which is why what is happening at other levels of politics is probably as significant. Although many were tempted to write off the Liberal Democrats when Mr Blair became leader, they have progressed steadily in terms of their practical influence - and not just in Parliament. In local government the party now claims just under 5,000 councillors (easily ahead of the Conservatives). More important is its ability to influence decision-making. Strictly speaking, few councils are ruled by coalitions, but in dozens no one party has overall control. According to the Liberal Democrats there are around 100 authorities where they have influence.
This may be a high watermark reached because of the acute unpopularity of the Tories. On the other hand the collapse of the Conservatives in local politics could create new opportunities. Either way, devolution to Scotland and Wales produces exciting opportunities for Mr Ashdown's party. In Scotland the prospects are particularly enticing because the Parliament will have real law-making and tax-raising powers. The electoral system (the Additional Member System) keeps constituencies but has a "top- up" element constructed on a proportionate basis to compensate for the imbalances produced by first-past-the-post voting.
No Labour campaign in recent memory has won 50 per cent of the vote in Scotland (where the presence of a fourth party, the Scottish Nationalists, divides the vote. As a result, a Lib-Lab administration is the odds-on favourite outcome. Links between leading Labour and Liberal Democrat politicians are good in Scotland, and the parties worked together on the devolution plans in the Scottish Constitutional Convention. The understanding was developed during the last year when Robin Cook, an advocate of proportional representation, pioneered pre-election co-operation on constitutional change between the two parties.
There are possibilities, too, in Europe. After some uncertainty Mr Blair has agreed to introduce proportional representation (under the regional list system) in time for the next European elections in 1999. The Liberal Democrats, who won 16 per cent of the vote but secured only two seats in 1994, stand to be significant beneficiaries. But that is not all. Fascinating political permutations are already being thought about, prompted by the fact that as little as 10 per cent of the vote in some regions could produce a seat. The Greens claim to have had discussions about constructing a "rainbow alliance" with left-wing rebel Labour MEPs. There are mutterings about pro-European Conservatives constructing their own party list; or even of the Tory party having two lists - one for pro-Europeans, the other for sceptics.
There may also be strain on the Conservative Party from two other directions. Electoral reform for the House of Commons itself is now a real possibility, and it poses a direct threat to the Tories' chances of regaining power in the near future. Support for electoral reform, which is mixed within the Labour Party, is strong among new Labour MPs, many of whom were not expected to win in May and who might find it difficult to hold their seats at the next election under the current rules. As power is devolved to institutions with at least some PR, so the Commons will start to look anomalous.
With Labour committed to holding a referendum on change in this Parliament, the voters may get a chance to give their verdict in 1999. Pro-reformers are confident of gaining some minority support among Conservatives; they believe constitutional reformers like Peter Temple-Morris could divide Tory ranks. One Liberal Democrat argued: "Whatever happens, the campaign for PR will include representation from Labour and the Tories as well as us."
An even worse prospect for the Tories' new leader, William Hague, is the possibility of a referendum on entry into the European single currency. That gained ground last week with reports of a new enthusiasm in the Cabinet for entry into the "second wave" of monetary union.
The consequence would be a split in the Conservative ranks. With Mr Hague committed to opposition to EMU during this Parliament, his opposition might succeed in peeling off the dwindling band of old, one-nation Tories and paint the rest of them as an English nationalist party.
The timing and outcome of these referendums depend on Mr Blair, who will make a judgment based on the political imperatives as the time approaches. But even without them politics is entering a new period of fluidity. For Mr Ashdown there are problems as well as opportunities. Not all the forces at play are pushing in the same direction. On a number of local councils Liberal Democrats are forging alliances with Conservatives, not Labour. In many areas Mr Blair's party is the clear rival. If Labour becomes unpopular mid-term there will be pressure to capitalise on anti-government votes.
Underlying Liberal Democrat concerns about co-operation lies the fear of being swallowed up by Mr Blair's new consensus. As Labour adopts traditional Liberal Democrat causes (such as modernisation of the constitution and operational independence for the Bank of England), so Mr Ashdown is left with less to differentiate his party from Labour. Some senior Labour strategists believe the Lib-Dems will split, with Mr Ashdown and his allies joining a coalition while some MPs and many activists concentrate on opposition and pavement politics.
The Liberal Democrats are trying hard to consolidate their position by fostering distinctive new policies to bolster their identity. To that end, Mr Ashdown announced the setting up of a new think-tank last week. His ideal is to get the best of both worlds by competing with Labour at some levels and co-operating at others. At the moment, however, co-operation is in the ascendancy, and with some justice, modernisers in both parties feel that history is on their side.
"These," said one last week with complete certainty, "are the first seismic shakes of a massive earthquake."