The chore is also an opportunity for Mr Blair to solve a number of outstanding problems, or at least apply some reassuring figures to those problems. Consider the simple beauty of the following series of key moves - Peter Mandelson becomes Secretary of State for Northern Ireland, Mo Mowlam moves to Health, and Frank Dobson is the official Labour nominee for the post of London mayor. If he did indeed achieve this blissful concatenation, Mr Blair could don his favoured leisure wear and go whistling off to Tuscany, contemplating the prospect of a settled autumn.
For a start, he would have managed a feat of personal importance to him: the early return to influence of Mr Mandelson. Peter in exile is like one of those characters in a soap-opera who leaves the story-line after a searing drama, with many tears shed and hearts broken, but somehow remains an indelible part of the show. We know that one day, like the eternally recurrent Frank Butcher in Eastenders or Brookside's turbulent son Barry Grant, he will return to the swirling plot of Westministers.
The ex-minister is very bad at doing nothing, hence his doomed attempts to launch comebacks in various forms earlier this year. This week, in a speech calling for closer European integration, he sought to establish himself as a prime mover on behalf of the embattled euro. Although he feels strongly about the issue, this does not strike me as a particularly prudent route back to the first rank.
Whatever one's view of the single currency, there is obvious uncertainty at the heart of the Government about how fast a sceptical population can be prodded into accepting Economic and Monetary Union. In a speech in Brussels this week, Mr Mandelson spoke of the need to "remake and win the case for Europe". Europe, by the way, is what EMU-philes now call the single currency in the hope that we will not notice the difference.
If Mr Mandelson does come in from the cold wearing the mantle of roving European envoy, built-in tension is guaranteed between his purposefully optimistic signals that Britain should and will join the euro zone and Mr Blair's more cautious assessment that it would be "daft" to join before the currency has been proved to work in Britain's interest.
Northern Ireland, on the other hand, would enable Mr Mandelson to effect a smooth re-entry to Cabinet in a post whose responsibilities and boundaries are clear-cut. Neither Gordon Brown nor John Prescott, the two heavyweights who are most sensitive about Mr Mandelson's come-back, would be unduly riled, since this particular office does not cut across their fiefdoms. True, it is still rather early for Peter's rehabilitation, as many a backbencher will point out at the slightest provocation. But Mr Mandelson, for all his flaws, is not just any other fallen minister. He is a shrewd and mature politician, capable of assimilating complex matters and acting decisively. New Labour has too few of those to let him languish around on the margins for too long.
Of course, it is not the prime purpose of the summer reshuffle to make Mr Mandelson happy. The idea is to re-balance the Government where it has become, or threatens to become, unstable or counter-productive. The label of failure is looming large over the Good Friday agreement. Whatever happens after tomorrow's decommissioning deadline, London's relations with the province will need a careful overhaul.
Unlike Mo Mowlam, Mr Mandelson does not come saddled with instinctive sentimental sympathies for the republicans. David Trimble, the moderate Unionist leader whose support is essential to the continuing quest for peace, respects him: Ms Mowlam would shed as few tears at leaving Stormont as the Unionists would at seeing the back of her.
The Prime Minister's initial instinct was to invent the post of party chairman and allow Mo's earthy charm to work its magic on those ranks of disaffected activists no other senior figure can ever quite find time to love. But Ms Mowlam, like anyone who has tasted real ministerial power, is deeply sceptical of trading it in for a mere party role.
That job might be better filled by Mr Prescott's ally, Ian McCartney, who could be charged with breathing some life into the moribund process of policy consultation with the party faithful. Ms Mowlam is dead set on Health.
For all the injunctions to think the unthinkable, Number 10 has not yet opened the Pandora's Box marked: "Future of the NHS". Like education, it is a public service badly in need of reform in structure and funding. Unlike education, the process has not even begun. So an honest job description would go as follows: "Must be jocular, with the common touch, sympathetic but non-committal when faced with public ire over the latest shortcoming. Requires the ability to present minor spending rises as humungously important funding rises and pass off fiddles with the waiting list figures without blushing."
Mr Dobson has performed this dramatically demanding role well. Ms Mowlam would excel at it. To be crude, and politics is a crude business, one can think of no better incumbent than a well-liked woman who has herself undergone cancer treatment and is thus a kind of walking embodiment of the successes of the service and of empathy with the suffering.
One other vacancy is likely to arise in the Cabinet Office: Jack Cunnigham has the gait of a dead man walking. He may well be replaced by Mr Blair's bouncy Number 10 confidant, Lord Falconer. Pat Hewitt, as the star pupil at the Treasury, should also be rewarded with a higher profile role, from which she can nanny us all to her earnest heart's content.
That leaves London, which sounds like small fry compared with the NHS and Northern Ireland, but is like a nagging backpain to Mr Blair. Unless he can find a figure behind whom the London Labour Party will unite, Ken Livingstone's campaign will continue to gather momentum.
Mr Dobson is the only candidate Mr Livingstone fears, since the whiskered Cockney grand-pappy does not fit readily into Ken's division of the mayoral race as a choice between himself - loveable outspoken populist - and "some ghastly toady".
But, Houston, we have a problem. Mr Dobson does not want to be the mayor. Mrs Dobson very definitely does not want Mr Dobson to take the job and is pressing her husband to accept retirement rather than touch it. Unless there is a change of heart in the Dobson household, this bet is off, the chain of dream solutions is fractured, and Mr Blair has another bout of brooding to do in Tuscany.Reuse content