I don't suppose David Cameron is looking forward to the publication of Lord Justice Leveson's report on the press next month. The Prime Minister squirmed through an uncomfortable day at the inquiry in the summer, when his close friendship with the former Sun and News of the World editor, Rebekah Brooks, was revealed in a series of texts and emails. But he must have breathed a sigh of relief last week when the inquiry did not order him to hand over another cache of messages between himself and Brooks, which is said to include embarrassing exchanges.

The fact that the inquiry hasn't demanded the emails is perplexing. It has rightly gone beyond the phone-hacking scandal to probe the relationship between editors and politicians, producing some of the most fascinating exchanges of the whole inquiry. Who could forget Cameron's chagrin as a text from Brooks promising a "country supper soon" was read out in the courtroom? Now it turns out there are many more messages between Cameron and Brooks, and between Cameron and his former communications chief Andy Coulson, but the Prime Minister believes he's entitled to withhold them, assured by lawyers that they're not "relevant".

I'm sure Cameron was very relieved indeed to get this advice. (Who paid for it, by the way? Earlier this year, the Cabinet Office told me that ministers were entitled to "legal support" for those parts of their evidence that relate to government business – but not for the period when they were in Opposition.) But it is surely up to Lord Leveson, not the Prime Minister, to decide which documents are relevant to the job he's been asked to do. Nor did Cameron help his case with a petulant performance in the House of Commons, where he refused to answer a parliamentary question on the emails from the Labour MP and phone- hacking victim Chris Bryant.

On Friday, the Prime Minister replied to a letter from Labour's deputy leader, Harriet Harman, and insisted that he had "cooperated fully with he inquiry and given them all the material, that they have asked for". But that isn't the point, as Bryant immediately pointed out. "It stinks of a cover-up," he said.

The question isn't what the inquiry asked for; it's about the prime minister's judgement and what qualifies him to make decisions about the relevance of material in which he's personally involved. If the messages consist only of innocuous observations about the delights of the Oxfordshire countryside, their disclosure would do no lasting harm. So the obvious inference, as with the Attorney General's decision last week to block publication of Prince Charles's letters to ministers, is that they're much more revealing than that.

Up to now, Lord Leveson has gone about his business with a polite but steely resolve, and I cannot imagine why he hasn't dispatched an officer of the court to Downing Street with an instruction to hand over the emails forthwith.

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