Cher walks into a London hotel room looking brilliantly Cher-like. She is wearing a leather jacket, tiny skirt, fishnets and big, buckled biker boots. She's got the blow-dried bouffant and pristine makeup of a pop star, but also the subtle swagger of a rock star – just witness the way she instantly dismisses the swarm of managers buzzing around her: "I hate talking with a thousand people around me."

Even more brilliant is how unintimidating Cher is. Up until a year ago that might have been a surprise, but when she joined Twitter in 2012, her public image was transformed. This, for instance, is her current Twitter biography:

Stand & B Counted or Sit & B Nothing. Don't Litter,Chew Gum,Walk Past Homeless PPL w/out Smile.DOESNT MATTER in 5 yrs IT DOESNT MATTER THERE'S ONLY LOVE&FEAR.

Her tweets are a riot of politics, bemusement, CAPS LOCK and emoji symbols. If Believe, the 1998 hit that practically invented the now ubiquitous pop vocoder vocal, made her seem robotic and remote, then Twitter has made her human and open again. Our chat today involves advice for Miley Cyrus, the perils of Grand Theft Auto and a lengthy story involving Salvador Dalí, an orgy and chocolate clocks.

Cher signed up to Twitter because a friend told her she'd be good at it. "I can't spell, my grammar is terrible, but I'm on it all the time because I enjoy it," she says, adding: "but sometimes even I go, you know what, shut the fuck up and go do something." I ask her about her emoji use and she grins. "One of the kids had it and I thought, I have to have that!" Her favourite is a ghost sticking its tongue out. "That's happy. That's me dancing, when I put that on."

Cher is 67 now, and has been dancing since her first single came out in 1965 – her recent album was her 25th solo effort. She left her wholesome Sonny & Cher image behind and shocked the public with a skimpy leotard in the If I Could Turn Back Time video way before Miley foam-fingered her way to a media storm. Cher is more qualified than most to talk about her, and she doesn't agree with Sinéad O'Connor's open letter warning Miley that the music industry "will prostitute you for all you are worth, and cleverly make you think it's what YOU wanted".

"Well, you're talking to someone who's done that kind of stuff, and I never did anything I didn't want to," Cher shrugs. "I was pushing an envelope that I wanted to push. I felt that looking the way I wanted to look and doing what I wanted to do made me more empowered. Sinéad O'Connor has a very different sense of herself than Miley Cyrus, so I don't think it's as dire as people are warning. Women have always been sex objects, and that's nothing new. They always will be."

The difference, you could argue, is that back in 1989, when Cher was straddling a cannon in the Turn Back Time video ("so tame in comparison to now"), the concept of a tween fanbase didn't exist. Should Miley and Rihanna, for example, care that their 10-year-old fans are watching them perform half-naked? "I don't think the naked part is the problem," she insists. "Unfortunately, what happens now is that kids are exposed to everything that's too old for them. You know, since the internet, since horrible PlayStation, you know, what's the name of it? Hijack car … that car hijack thing? I think maybe that is worse than seeing Miley Cyrus on a wrecking ball. Also, she's trying to break the Disney stuff. When you make such an image, you have to hit it with a hammer. Or a sledgehammer!"

At the time of Miley's MTV performance, Cher was drily scathing: "I don't think it was her best effort." But she says now that she'd been doing interviews all day, "then somehow, I started liking the sound of my own voice pontificating. And finally I thought, this is shit, you know? Get off this kid. It was one performance. It turned out to be the greatest stunt of all time 'cos people are still talking about it. She doesn't give a shit what I think, anyhow."

When Cher finally called time on Sonny & Cher, he told her: "America will hate you and you won't have a job." Little wonder, then, that she too was desperate to break away from her wholesome, unrealistic image. "When I left Sonny, I wanted to break Sonny & Cher. I wasn't them, and I wanted to be me. It was such bullshit. I never really did anything."

Cher is at her most angry when talking politics. She thinks what is happening to women in the US is "terrible … we've lost all of our rights. They couldn't do it federally, so they're taking it away state by state. I don't recognise my country," she says, sadly. "That's not funny to me at all. I don't know what the fuck happened. These people [the Tea Party, whom she regularly refers to on Twitter as "t-hadists"] hijacked it, and they're bringing down the prestige, and the dollar, for whatever it's worth."

Sadly, President Cher is not an option. "Oh God no, absolutely not. I'd be the worst person in the world. I have a horrible temper when it comes to that sort of stuff. I'd go around telling everyone what I thought, and being a politician is about not saying anything." She has first-hand experience of power, she reminds me: Sonny was a Republican congressman. "He wasn't a great statesman, but when people got into an argument, Sonny could bring everyone around the table, and say, come on, let's play cards, let's have a drink, let's work this out. Since the Tea Party, and since the Republicans have gone so far to the right, you can't even deal with them. Who'd want to?"

The mood has darkened. We've decided that politics are a wasteland and the future is looking grim. So it seems a good time to ask if she feels responsible for the Auto-Tune sound that has defined the past 15 years of pop music, thanks to Believe. "I was the first!" she says. "Believe was such a horrible song, and Mark [Taylor, who co-produced] and I fought over it constantly. He kept saying, sing it better, and I kept saying, I'm singing it as good as I can. Eventually I said, fuck you, if you want it better, get another singer, and I stormed out." Perhaps she's right about that political career, after all.

At this point, GMTV makes a surprise entry into pop folklore. At the time, Cher was living in the Docklands in London. "Do you know Roachford? Beautiful young boy … anyway, I was watching Good Morning Britain or whatever, and he was on, singing through a vocoder." She took the idea into the studio the next day, and Believe finally came to life ("I was over the moon.") A decade and a half later, you can still hear that sound all over the charts. "It's strange that an artist so old can come up with something that an artist so young is still doing," she muses. Her new album, Closer to the Truth, still makes use of it throughout its bangers and ballads, as if reminding everyone where it came from.

Believe was a smash in gay clubs, but Cher has had gay fans, and drag impersonators, since the early days. "Gay guys like a certain kind of woman," she explains. "They like a flamboyant woman that's broken. They like a balls-to-the-wall woman, motherly but not; sexual but not. Gay guys are like this: they either love you or they don't even know you're on the planet. Once you have them, you have them." I suggest that she's got a lesbian fanbase, too: she's been refreshingly frank about coming to terms with the sexuality of her transgender son, Chaz, and her first film role had her playing Meryl Streep's lesbian roommate in Silkwood, but she shakes her head. "Not in the same degree. I think my gay boys, that's a big part of my world."

Cher's last movie was Burlesque in 2010, in which she played the hard-bitten boss of a dance troupe. In the 80s, Cher's acting career was big news: she carved a niche as a tough mother with a heart in Face and Mermaids, and won an Oscar in 1988 for the quick-witted Moonstruck, in which she played a no-nonsense New Yorker falling in love over opera. Why did the acting dry up? "I'm in a strange place right now," she says, making a small gesture towards her face. "I'm too old to be young and I'm too young to be old, so I have to be used creatively. In Burlesque, which was horrible, I had no love interest, I was running this [troupe], that's who I was."

Cher's right about Burlesque – an overlong potboiler that also starred Christina Aguilera, it wasn't even camp enough to be fun. "It could have been a much better film. It was always sad that it was not a good film," she says, pinning the blame on its writer/director, Steve Antin. "Terrible director! Really terrible director. And really terrible script. I remember him saying to me, I don't care about what you say, I just want to shoot the dance numbers. Had it been shorter, it would have squeaked by and been a really good popcorn movie." She'd like to think she can act again, though. "I would like to do that, and I will do it, you know."

I want to talk to Cher for hours, about everything, but her PR and a woman called Susan, who works for her label, have entered the room, signalling that I have time for just one more question. A friend of mine, a superfan, told me I must ask her about Salvador Dalí. So I do. Is it true that Dalí gave her a vibrator? "Whooooo?" she shouts. Salvador Dalí, I say. She lets out a long, high-pitched howl of recognition, gets comfortable, and tells the three of us this.

"This is a complicated story. So, Salvador invited me and Francis Coppola and Sonny and my girlfriend Joey to dinner. And so we got to the apartment and they'd been having an orgy in the other room. People were in different stages of undress, but mostly dressed. They were staggering around and speaking French, just crazy, you know? So I have my hand on the chair and I see something in the crack, and it's a beautiful, painted rubber fish. Just fabulous. It has this little remote-control handset, and I'm playing with it, and the tail is going back and forth, and I'm thinking it's a child's toy. So I said to Salvador: 'This is really funny.' And he said [she puts on a deep, comedy Spanish voice]: 'It's wonderful when you place it on your clitoris.'" The room explodes.

"Now, Sonny and Francis, these little boys, are sitting there unable to control themselves, and they start eating the chocolate on the table. This happened to be these clocks that Salvador had made for decoration, and Francis and Sonny got so nervous they started eating them, these fabulous candy clocks."

The six of them walked to the restaurant, but alas, their dinner date with Dalí was not to be. "Ultra Violet [Dalí's assistant] was there, and she kept rubbing me with her cane, and I thought [chuckles], I'm going to break this cane across this bitch's head. All of sudden, Salvador and Ultra Violet got up, moved to another table and sat down. We became hysterical. We were crying. And they sat there and had their dinner, and we sat there and had our dinner." So she didn't keep the vibrator? "No, I didn't keep it!" She sits back on the sofa, and shouts across the room. "I bet you haven't heard that one before, Susan!"