On the new season of “The Crown,” newly elected Prime Minister Margaret Thatcher and Queen Elizabeth II seem to get on well enough during their first meeting.

Thatcher promptly curtsies, almost to the ground. The Queen correctly predicts Thatcher’s cabinet.

Later that evening, Thatcher gives her husband, Denis, the play-by-play.

“Smart cookie,” Thatcher’s husband says.

“Quite different than I imagined. More interesting and informed, with a commendable appetite for work,” Thatcher replies. “I left thinking we might work very well together.”

Denis isn’t entirely convinced.

“Two menopausal women,” he says. “That’ll be a smooth ride.”

He was right, in a sense. And wrong.

There was, indeed, tension between the two women during the 11 years Thatcher reigned as prime minister while the queen reigned as the sovereign — an extraordinary period when one of the world’s superpowers was led by two supremely powerful women. But hormones weren’t the cause of their icy relationship.

For starters, they both had daddy issues.

“Both women were shaped by intransigent men who disliked compromise,” wrote historian Dean Palmer in “Queen and Mrs. Thatcher: An Inconvenient Relationship.” “Their fathers were born in an era when class and social position were fixed at birth and remained unchangeable until you died. The two men found it impossible to escape their respective birthrights.”

Thatcher’s father, Alfred, was a shopkeeper and alderman. The queen’s father was King George VI — forced onto the throne by the extraordinary abdication of his brother to marry an American divorcée. Alfred seemed to never stop working. The king liked to hunt.

“With Alfred, life was about pulling yourself up by your boot straps and making something of yourself,” Palmer wrote. “By contrast, the queen’s father, George VI, was determined to resist change in whatever shape it might appear; for him, maintaining the status quo was the highest virtue. These paternal philosophies would stick like glue to their respective daughters. To understand both women, you must understand the fathers.”

“The Crown,” a fictional series based on fact, explores these tensions throughout season four.

In the second episode, the queen invites Thatcher, played by Gillian Anderson, and her husband to the family’s Balmoral estate in Scotland. A servant leads the couple to their bedrooms — plural.

“Two bedrooms?” Thatcher asks.

“It’s all very odd,” her husband replies.

“Are we allowed to sleep in one bed?” Thatcher says.

“I shall go check with the protocol sheet,” Denis replies.

A moment later, a servant interrupts the couple.

“I couldn’t help noticing but you didn’t bring any outdoor shoes?” the servant says.

The prime minister is confused. The servant leaves.

“What a strange thing to say,” Thatcher says.

It soon becomes clear why Thatcher’s lack of outdoor shoes is royally problematic. The queen invites her to go stalking in the wilderness. Thatcher arrives in a formal blue suit and heels. On a drive to the stalking point, the women engage in a conversation that sums up their differences.

(It is not known if this exact conversation took place. Artistic license, etc., etc.)

“I am afraid we are all mad stalkers,” the queen says. “It was how I spent some of the happiest times with my father, King George. He taught me everything.”

Thatcher says her father taught her things, too.

“We worked,” she says. “Work was our play. I worked with him in our shop. As an alderman, he took me everywhere. I watched as he wrote his speeches and listened as he rehearsed and delivered them. It was my political baptism.”

The queen smiles.

“How lovely for you both,” she says.

You will not be surprised to learn that the stalking did not go well. And neither, according to the media and historians, did their ongoing relationship.

Palmer quotes William Whitelaw, Thatcher’s deputy party leader, as saying: “Throughout their weekly meetings, over the years, the two women maintained a rigid formality. The ice never broke. Margaret would have expected the queen to make the first move, and that never happened.”

But neither woman ever degraded the other in public, always keeping up appearances for the good of the nation. And Thatcher downplayed any tensions.

“Although the press could not resist the temptation to suggest disputes between the Palace and Downing Street, I always found the Queen’s attitude towards the work of the Government absolutely correct,” Thatcher wrote in her autobiography. “Of course, stories of clashes between ‘two powerful women’ were just too good not to make up.”

After Thatcher died in 2013, the queen’s press secretary released a terse but respectful statement: “The Queen was sad to hear the news of the death of Baroness Thatcher. Her Majesty will be sending a private message of sympathy to the family.”

Monarchs do not typically show up at the funerals of commoners. But in a move that stunned palace observers, the queen broke protocol and attended Thatcher’s.

“The only other time a reigning monarch has attended the final farewell of a Prime Minister was in 1965,” the Daily Mail wrote, “when the Queen joined the congregation for the funeral of Winston Churchill.”

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