Can a Labour prime minister get on well with a Republican US president? Or a Conservative PM with a Democrat in the White House?

The short answer is yes, absolutely.

There are plenty of examples of a good relationship and close bond between a Labour prime minister and Republican president. And vice versa.

Indeed, some prime ministers and presidents from seemingly opposing political parties have bonded for the simplest or most trivial reasons. Cigars, toothpaste and burgers, for example.

And it’s not always rosy between prime ministers and presidents of the two sister parties. There have been some big fallings out: over Suez, Vietnam and the Caribbean island of Grenada.

But never has a British Labour prime minister faced such special challenges in maintaining the “special relationship” with a Republican president as Sir Keir Starmer does right now.

It’s not just policy differences – on issues such as trade tariffs, Ukraine, Israel, defence spending, Brexit and climate change – that divide Downing Street and the White House right now.

Never before has an incoming president faced such a tirade of brutal insults from senior members of a UK government like those hurled at him by leading members of Sir Keir’s cabinet.

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He’s a “racist KKK and neo-Nazi sympathiser”, (David Lammy, 2017), an “odious, sad little man”, (Wes Streeting, 2017) and “a racist misogynistic, self-confessed groper”, (Ed Miliband, 2018). And that’s just a sample.

That’s not all. Last month, the Republican Party filed a legal complaint after almost 100 Labour Party aides flew to the US to campaign for Kamala Harris, alleging “blatant foreign interference” in the presidential election.

Critics, led by the new Tory leader Kemi Badenoch, have accused Sir Keir and his party of playing student politics by picking a fight with the most powerful man in the world. And someone who’s notoriously vindictive.

It was all very different 80 years ago (critics would also say that political leaders were real statesmen back then).

The phrase “special relationship”, describing the alliance between the UK and US, was first used by Winston Churchill in a speech in Missouri in 1946, in which he also coined the phrase “the Iron Curtain”.

That speech was introduced by president Harry Truman, a Democrat, with whom Churchill had attended the Potsdam Conference in 1945 to negotiate the terms of the end of the Second World War.

They were close friends and would write handwritten letters to each other and addressed one another as Harry and Winston. Truman was also the only US president to visit Churchill at Chartwell, his family home.

Churchill also had a close relationship with another Democrat president, Franklin D Roosevelt. Their close bond during the Second World War was described as a friendship that saved the world.

One reason they got on famously was that they were both renowned cigar smokers. Like Churchill, Roosevelt’s cigar smoking was a widely reported part of his public persona after he became president.

But after Churchill’s bromances with Democrat presidents, his Conservative successor Anthony Eden fell out badly with the Republican president Dwight Eisenhower over the Suez Crisis in the mid-1950s.

And it was a Conservative prime minister and a Democrat president with seemingly nothing in common, the stuffy and diffident Harold Macmillan and the charismatic John F Kennedy, that repaired the damage.

“Between them they had rescued the special relationship after the rupture of the Suez Crisis, and done so at a time of uniquely high tensions around the world,” wrote British author Christopher Sandford in Harold And Jack, The Remarkable Friendship Of Prime Minister Macmillan And President Kennedy.

It was the early 1960s and these were dangerous times, rather like now, of course. Back then it was the Berlin Wall, the Cuban missile crisis and threat of nuclear weapons.

“Through it all, the two leaders had exchanged not only formal messages but also a steady flow of handwritten notes, Christmas and birthday cards, congratulations, and, on occasion, condolences,” Sandford wrote.

But it was a relationship abruptly cut short in 1963, by “supermac’s” demise caused by the John Profumo sex scandal and then JFK’s assassination in Dallas just a month later.

“Like many of those who came into the Kennedys’ orbit,” the Washington Post wrote, “Macmillan was enchanted by Jacqueline Kennedy, and she seems to have happily entered into a father-daughter relationship with him that lasted long after her husband’s assassination.”

After Kennedy, the so-called “special relationship” cooled once again during the tenure of Labour’s Harold Wilson and Democrat Lyndon Johnson, when Wilson rejected pressure from Johnson to send British troops to Vietnam.

And even though Margaret Thatcher and Ronald Reagan were ideological soulmates, Thatcher was furious when she wasn’t consulted before the Americans invaded Grenada in 1983 to topple a Marxist regime.

Even worse, according to Thatcher allies, a year earlier Reagan had stayed neutral during the Falklands war. Reagan said he couldn’t understand why two US allies were arguing over “that little ice-cold bunch of land down there”.

Long before the accusations of Starmer’s Labour meddling in the Trump-Harris election, the Tories were accused of dirty tricks in the Bill Clinton-George HW Bush presidential election of 1992.

During the campaign the Home Office checked immigration nationality records to see whether Clinton applied for British citizenship while a student at Oxford University to escape the Vietnam draft. It wasn’t true.

Then prime minister John Major issued a grovelling public apology and Clinton was forgiving. In 1994 the “special relationship” received a huge boost when the president took Major to the home in Pittsburgh where his grandfather and father lived and worked.

Then it was back to Washington where Major became the first foreign leader to stay overnight in the Clinton White House. But as well as the flattery, the pair worked closely in the early stages of the Northern Ireland peace process.

Clinton’s political soulmate, of course, was Tony Blair. They were as close as Reagan and Thatcher. But it was with the Republican George W Bush that Labour’s Blair embarked on the defining mission of his premiership, the Iraq war.

George “Dubya” Bush had defeated Clinton’s vice president Al Gore in the bitterly contested presidential election of 2000 and in early 2001 he entertained Blair at Camp David. It was to prove to be a historic encounter.

“He’s a pretty charming guy,” the president gushed at their news conference. “He put the charm offensive on me.” How many times have we heard that said about Tony Blair?

Then it got deeply personal. They were asked if they’d found something in their talks that they had in common. “Well, we both use Colgate toothpaste,” the president replied.

Quick as a flash, an embarrassed Mr Blair intervened: “They’re going to wonder how you know that, George.”

The war was the turning point of Blair’s decade in Number 10. He was branded a liar over claims about Saddam Hussein’s “weapons of mass destruction”, he was vilified by the Labour left and it was the beginning of the end for him.

Some years later, the Tory prime minister sometimes called the “heir to Blair”, David Cameron, bonded over burgers with the Democrat president Barack Obama, serving a BBQ lunch to military families in the Downing Street garden.

They seemed unlikely allies: Obama the first African-American president and Cameron the 19th old Etonian prime minister. It was claimed they had a “transatlantic bromance” in office.

The two leaders were often pictured together playing ping-pong or golf, eating burgers or watching a basketball game. “Yes, he sometimes calls me bro,” Cameron once said of president Obama.

Cameron even persuaded Obama to help the Remain campaign in the 2016 Brexit referendum, when he claimed the UK would be “at the back of the queue” on trade deals with the US if it left the EU.

Which brings us, neatly, to Sir Keir and president-elect Trump and the prime minister’s hopes of building a special relationship.

On the plus side, the president likes the UK – his mother was Scottish and he owns two golf courses in Scotland. And we’re told by Sir Keir that the dinner at Trump Tower in September went well. The mouthy Mr Lammy admitted he was even offered a second portion of chicken. “He was very gracious,” he claimed.

On the other hand, neither the prime minister nor the president smoke cigars, like Churchill and Roosevelt did. We’re not sure which toothpaste they use, unlike Bush and Blair, either.

And while the president obviously likes burgers – he famously flipped them in a McDonald’s during the election campaign – and steak, well done, with ketchup, Sir Keir is vegetarian, though he does eat fish.

But if even a stuffy old toff like Harold Macmillan can get on well with the flamboyant JFK and glamorous Jackie Onassis, there’s hope for Sir Keir and that much-vaunted “special relationship”.