A sexy Cyndi Lauper costume can be created with a lace skirt, fishnet leggings, tube top, fingerless gloves, and a lopsided big wig. Whether you’re going to your high school reunion, to an 80’s themed party, or to a costume party, a Cyndi Lauper costume would be perfect. Other 80’s Halloween costume icon ideas, include the members of the band Kiss, Blondie, Billy Idol, Madonna, and of course, the white white gloved Michael Jackson.

It’s hard to imagine Madonna making such a gesture, let alone watching approvingly as the cannoli is swallowed. It may be unfair to still be comparing Lauper to Madonna 25 years after the pop rivalry that resulted in the arguably more talented Lauper being relegated to runner-up, but the similarities make it hard to avoid. Both are Italian-American Catholics whose personalities engulf their music; both are in their third decade in a business that often measures a career in months; and both have influenced subsequent generations of singers. Gwen Stefani and Alison Goldfrapp certainly seem to have taken Lauper’s full-color kookiness as their model.

There is also a new point of comparison: Lauper’s first album of new material since 1997, Bring Ya to the Brink, is a back-to-the-disco record that has things in common (including the use of hip young producers) with Madonna’s recent return to her early 1980s sound. The difference is that the crackling Bring Ya to the Brink sounds like it was made not with platinum discs in mind but to sate Lauper’s massive appetite for musical experimentation. That’s just as well; its first single, Into the Nightlife, sold well, but the album, despite thrilling the critics, hasn’t had the same success. Is she bothered? Well, one of her favorite expressions is "I don’t give a hoot!", and she does seem to mean it.

There’s a lullaby sweetness to her speaking voice that contrasts markedly with both her powerhouse singing style and the New York accent that coats every word. Years ago, as a guest on Terry Wogan’s TV show, she was ribbed by Wogan about the accent and, supremely straight-faced, told him: "Most people think I’m cockney." It transpires that she was being serious. "People do say I sound English sometimes," she says. "At home, sometimes they say that." Home is still New York, where they clearly have a novel perception of the English accent.

But maybe I’ve misunderstood: Lauper’s thoughts aren’t always easy to follow, sometimes emerging as a stream of images that don’t bear much relation to the question she’s meant to be answering. Asked how she’d hooked up with Basement Jaxx, who co-wrote the album’s fabulously spaced-out track Rocking Chair, she instead tells a story about taking the Eurostar to London from Paris. "The train goes under the water, so you don’t see the white cliffs of Dover! When we went through the tunnel, I was sitting there, quietly singing The White Cliffs of Dover to myself." She then describes what she’d been wearing on the train – lots of enveloping layers and a fedora – and then sighs, saying: "When you’re young and walk down the street and look eccentric, that’s cute. But when you’re older, they’re scared of you. Get to the other side of the street! The difference between the way you look and the way you think you look! I thought I looked like Greta Garbo."

There’s nothing eccentric about her appearance today. She’s downright glamorous: stylishly angular platinum hair, black jeans, black sweater and fierce, high-heeled black boots. She looks years younger than 55, and is very pleased when I say so. "Ageism? There’s ageism in everything. I don’t give a hoot. It isn’t what other people think; it’s what you think. But it’s hard to come to terms with getting older. I admire people like Vivienne Westwood." And Madonna? "She’s fantastic. She’s like Wonder Woman."

She’s keen to discuss Bring Ya to the Brink, written and recorded in Europe. "I wanted to experience writing in a different place. I never went to Europe to write before. I didn’t go as a famous person, I went as a writer and stayed at a hole-in-the-wall hotel in Chelsea, and I could come and go without being noticed. I went to Brixton [to work with Basement Jaxx] and that was so great."

Despite having sold around 25m records – including one, Girls Just Want to Have Fun, that’s considered the definitive pop moment of its year, if not the early 1980s – Lauper doesn’t relish the limelight, and worries about its effect on Declyn, 10. "He was two when he first realized I was famous," she says. "It’s hard for him. People look at him, and he feels the pressure of being my son. Someone came up to me in the street and kept talking and talking. Declyn said afterwards, ‘Who are you?’ And I said, ‘I’m a mommy.’ And he said, ‘No, you’re not – you’re Cyndi Lauper!’" She searches through her handbag and pulls out photos of a long-haired little boy wearing a hockey mask. She laughs: "I’m a hockey mom."

This leads to a brief discussion of the moment’s other renowned hockey mom, Sarah Palin, for whom Lauper has mixed feelings: "She gives a good soundbite. It’s great McCain picked a woman, but it’s sad he picked one who’s unqualified." On her blog, she has reproduced an essay by Gloria Steinem, the US feminist icon, that rues the choice of Palin, and the site provides a link to register to vote. On her True Colors tour this summer, she banged home this same message, telling audiences: "It’s more important to vote for the American president than the American Idol."

You don’t do a bad soundbite yourself, I say. She looks dubious, as if I’ve insulted her, then decides no offense was intended. She’s a sensitive person, she says, who’s always felt herself an outsider, "and not through choice. I’ve always been the outsider looking in." So the title of her multi-platinum 1983 debut album, She’s So Unusual, was apt? "It wasn’t a joke," she says.

The back cover of Bring Ya to the Brink shows Lauper standing outside a suburban house, gazing into a window. This photo, inspired by a painting she did while studying art as a teenager in Vermont, has only one interpretation: exclusion. Despite the professional success, the 17-year marriage to Law & Order actor David Thornton, and the warm reviews that greeted Bring Ya to the Brink, she still sees herself as the odd one out. Not that she’s bothered.

"Do people know who I really am, what I really do?" she says. "No. Do I care? No".