LONDON — It is the most important, the most inviolable rule of Britain’s modern monarchy: The queen shall not muck about in politics.

Her Majesty doesn’t take sides. Forget all the royal gossip between squabbling duchesses. On matters of state, Her Royal Highness stays impartial.

But now it appears that Queen Elizabeth II may have gotten her brushed white cotton gloves a little dirty?

Former prime minister David Cameron revealed Thursday that he, ahem, suggested to the queen’s courtiers that she could and should try to very, very subtly influence the outcome of the 2014 Scottish independence referendum.

With a wink or word — or even an arched eyebrow.

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And this, according to Cameron, the queen did.

An unnamed source told the BBC that Cameron’s comments have led to “an amount of displeasure” at the palace. A spokeswoman for Buckingham Palace told The Washington Post, “It’s not something we are making any comment on.”

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The revelation appears in a BBC series about Cameron’s life, to begin broadcast Thursday and tied to the release of his new memoirs, “For the Record,” which are surprisingly dishy.

The date is September 2014. Cameron and his wife are enjoying themselves at the queen’s beloved Balmoral estate in the Scottish Highlands, when a new poll rocks his world: The Scottish independence vote is surging.

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The survey hit him “like a blow to the solar plexus,” creating “a mounting sense of panic,” Cameron recalled — panic at the thought he would be the prime minister to preside over the crack up the United Kingdom.

And so, Cameron said, “I remember conversations I had with my private secretary, and he had with the queen’s private secretary, and I had with the queen’s private secretary, not asking for anything that would be in any way improper or unconstitutional, but just a raising of the eyebrow — even, you know, a quarter of an inch — we thought would make a difference.”

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Lo, just a week later, in the courtyard of Crathie Kirk, the church where the queen attends Sunday services while at Balmoral, a well-wisher disclosed that Her Highness told her, “Well, I hope people will think very carefully about the future.”

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Naturally, the media went nuts — as the words were seen as a coded message, a warning — and so became a major debate point in the campaign.

Cameron confessed to the BBC that the queen’s remarks, limited and opaque though they may have been, “helped to put a slightly different perception on things.”

Scotland rejected independence by a vote of 55 percent to 45 percent. The kingdom was saved. Cameron was “blissfully happy,” and so, he suggests, was the queen. A few weeks after the vote, Cameron was caught on a live microphone saying the queen “purred down the phone” when she was told about the result.

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But these latest comments weren’t just accidentally overheard. He made them willingly to the state broadcaster. Cameron’s indiscretion has left some palace watchers gobsmacked.

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A prime minister and the queen usually meet weekly to discuss matters large and small. But prime ministers are not supposed to dish.

“The first rule of the relationship between the prime minister and the queen is that you never, ever talk about the relationship between the PM and the queen,” observed BBC royal correspondent Jonny Dymond.

“It is difficult to imagine anything other than horror in the Palace at David Cameron’s revelations,” Dymond wrote. “Not just because he has broken the first rule. But because he has made it painfully clear that in 2014 he used the Queen for his own political purposes. And that she and her advisers thought that was okay.”

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Alex Salmond, who resigned as Scotland’s first minister in the wake of the 2014 referendum result, tweeted Thursday that Cameron’s actions were “improper” and an indication of how “desperate” the former prime minister was in the final states of the campaign.

On BBC radio Thursday morning, Cameron was asked to provide more details. He demurred.

“I’m sure that some people would think, possibly even me, that I’ve already said perhaps a little bit too much,” the former prime minister said.