Eileen Atkins is the only Dame of the English theater who hasn’t been invited to lunch with the Queen at Sandringham House. All the other Dames — Judi (Dench), Maggie (Smith), Diana (Rigg) and Joan (Plowright) — have had the pleasure of Her Majesty’s grilled Dover sole, but not Atkins. Curious, she asked a friend who looks after the Queen’s horses why that invitation has yet to arrive.

“He looked at me and said, ‘Bit of a loose cannon, you are, Eileen.’ ” She laughs. “Well, I suppose that’s true.”

Atkins, now 85 and giving one of her customary brilliant performances in Broadway’s “The Height of the Storm,” has always been candid. In 1971, she told Alfred Hitchcock he was a misogynist. The great director wanted her to appear in “Frenzy,” about a serial killer who rapes his victims and then strangles them with his tie. Atkins read the script and told her agent to say no, because she found it offensive. During rehearsals for a play, she got a call from Hitchcock.

“I hear you’re calling me a misogynist, and won’t do my film,” he said.

“Well, you are and I won’t,” she replied, and went back to rehearsal.

“Look,” she says over a lunch of salad and sardines in the West Village, “he’s made some of the greatest films ever, and so I feel a little terrible about it. But I still stand by opinion that ‘Frenzy’ is a bad piece of work.”

Don’t get her started on the #MeToo era. “My agent has made me promise I won’t talk about it,” she says. (“Bit of a loose cannon, you are Eileen.”) But she does say that one of the most important men in her life was in love with her when she was a teenager, and he was in his 30s. A drama teacher, he taught Atkins, a working-class kid with a Cockney accent, “how to speak properly,” she says. He was a defrocked priest who had married and was divorced by the time they met. “He behaved impeccably to me,” she says, “and he arranged everything for me at the beginning of my career. It was only afterward, when I was older, that he told me he loved me.”

Atkins recalls only one run-in with someone who behaved improperly. When she was 19, she went to a director’s apartment to discuss roles in his theater company. “He whipped open the door and said, ‘Ethel’s in the bath, my dear, give us a little kissy!’ Ethel was his wife. He held me against the door and slobbered a kiss on me. I said, ‘No,’ and he knew straight away I was not going to play, and he never came near me again.”

In Florian Zeller’s “Height of the Storm,” Atkins plays the wife of a famous, now forgetful novelist (Jonathan Pryce), who can no longer function without her. Zeller plays with time and memory, and it’s not always clear who’s alive and who may be dead. Audiences have been divided: I think it’s terrific, but Atkins says, “I have friends who hate it.”

She’s worked out her own interpretation, but says she can’t discuss it with Pryce or her director, Jonathan Kent.

“They’re obsessed with keeping it loose,” she says, “and not trying to figure out what may or may not be happening in a scene. But I try to get across my point of view.”

Atkins and Glenda Jackson may be the last of their generation of actresses who can still deliver eight carefully wrought performances a week. (Jackson did “King Lear” on Broadway last spring.)

Plowright retired in 2014, after going blind. Dench suffers from macular degeneration and hasn’t appeared onstage since “The Winter’s Tale” in 2015. Vanessa Redgrave struggled with her lines during “The Inheritance” in London and declined to reprise the role in New York. As for Maggie Smith: “She’d die if she heard you say she can’t do eight performances a week,” Atkins says. True, Smith won raves as Joseph Goebbels’ secretary in “A German Life.” But she, too, declined to come to New York — and sources say she’s in no rush to do another play.

Atkins says Smith did “A German Life” because “otherwise, she would have gone down in history as ‘Harry Potter’ and ‘Downton Abbey’ and she would have wanted to kill herself,” she says, laughing. “We have our standards.”

Atkins says her next play will be Amy Herzog’s “4000 Miles” in London, though she wonders how much more stage time she has left. “I saw Edith Evans when she was still wonderful, but I also saw her in ‘Hay Fever’ when she was over the hill,” she says. “That’s a big worry for me. I told my agent, ‘You’ve got to tell me when I’ve lost it.’ And then I will stop.”

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