The sort of behaviour that originally excited them was buying a Porsche with her earnings before she was old enough to drive one, but her marriage, when in her twenties, to a lawyer 30 years her senior aroused slavering interest. As did her subsequent marriage - after her husband died of cancer - to the much-married Andre Previn.
What arouses this prurience? The answer lies in the way her repeated gravitation to father-figures - starting with the flamboyant conductor Herbert von Karajan - is coupled with her statuesque beauty and the immaculate perfection of her art.
Critics tie themselves in knots trying to decide if her playing is too perfect for its own good, or not deeply enough felt, but the fact remains that she played like a goddess when she first hit the headlines at 13, and she plays like one still: wonderfully responsive to the grandeur of Beethoven, to the drama of Vivaldi, and the exhilaration of Stravinsky, and all with the purest eloquence of tone. She thinks big and bold, and she's a staunch promoter of new music.
She is now embarking on an ambitious Mozart project: recording and performing his major works for violin, starting the week after next with three gruelling nights at the Barbican. Mozart, she says, has always been present on a daily basis in her life. "I've never stopped thinking about him, even when I was playing contemporary repertoire, and I've always been trying out new ways to get closer to him."
When she started lessons, it all seemed too easy until her teacher, the redoubtable Aida Stucki, showed her that playing Mozart "wouldn't work simply in terms of playing him nicely". Stucki introduced Mutter to the Central European style of playing that she now exemplifies: it's the antithesis to that brash efficiency that characterises the stars from the Juilliard.
So which violinists have got Mozart right? Arthur Grumiaux and Isaac Stern, Mutter replies, but, since those greats passed on, nobody else. "Mozart's music is like an X-ray of your soul - it shows what is there, and what isn't. Yet he isn't taken very seriously by the younger generation."
In 1978, Mutter recorded two Mozart concertos: how does she feel about re-recording them? "A lot has happened in my life over the last 30 years - that alone seems reason enough to try a second time." Then she makes an observation that should be branded on every young prodigy's T-shirt. "What comes out when you are 12 does so purely by instinct. And if you continue to be guided by instinct, it's not enough; you just repeat yourself. You can only grow artistically if instinct is combined with a curious mind."
Part of my homework before this interview has been comparing her recent recordings with those she made two or three decades ago, and the point is well made. The 13-year-old Mutter delivered a flawless performance of Mozart's Concerto KV218, but compared with the new version out this week from Deutsche Grammophon, the difference is striking: her goal with the former seems to have been perfection per se, but now you feel she wants to take risks.
That's doubtless because she is taking risks: she has dispensed with a conductor, and is leading the band herself. "But I'm not pretending to be a conductor. I've never studied it." Then, after a pause: "But I am a leader - partly because it's my character, and partly because I know precisely what I want from the score, and know how to explain it to an orchestra."
Her intention was to record these concerti with the Vienna Philharmonic, and she insists the dry runs went fine: "The players came forward with some wonderful ideas. We had many discussions, and my ideas initially met some resistance, but I think I was able to convince them."
Well, word has it that that resistance - from the most misogynistic band in the world - was more than she could handle; the orchestra with whom she has now recorded these works is our own London Philharmonic, which she praises to the skies. They are "very modern" players, she says. "Some orchestras sound velvety, but the LPO are more like a Porsche."
Total assurance seems to mark everything this woman does. At 41, with two children from her first marriage for whom she ring-fences large amounts of time, she's busy with a host of altruistic causes. She has supported the building of a boys' orphanage in Romania, and is now helping build one for girls.
Meanwhile, the foundation she has set up for young string players is going full tilt. The students she accepts have all the necessities provided, as well as more delicately calibrated care. One student that she brought from Beijing had problems getting his head around the German repertoire. "So I climbed mountains with him, made him read German authors and took him to Leipzig to see Bach's Thomaskirche."
She has also persuaded the Bavarian state government to test out her educational ideas in 100 kindergartens. "To start playing at five or six isn't early - not if you are going to do it seriously." She presumably means gifted children? "I don't think there's any such thing as an ungifted child. Singing and music-making are the most basic human activities."
So where next? Playing concerts to promote the Mozart project will occupy all her energy for the next year: "I didn't know how engulfing it would be, or how interesting." Thereafter, we can expect more action on the Previn front, notably with a concerto he's written for a Slovenian double-bass player in her foundation, with her violin as the counterweight. "Andre is one of the very few composers who delivers on time, because he's a musician, and he knows what it feels like to receive the music two hours before you have to run on stage."
Does she ever plan to slow down? "My life will always be in high gear, even when it doesn't seem to be." What about those Porsches? "I'm repressing my desire for one at present - I want to set a careful example to my children. But when they've grown up, I'll get one again."
Mutter's Mozart, Barbican, London EC2 (020-7638 8891), 19, 20, 21 September. 'Mozart: the Violin Concertos, and Sinfonia Concertante, Anne-Sophie Mutter with Yuri Bashmet' is released by Deutsche GrammophonReuse content