Julia Greenberg was in a friend’s car, driving through New England, when a playful ballad about a Hollywood down-and-out came on the stereo. Who is this, she wondered. “She sounded so wholly unique,” Greenberg says, 15 years later, on a video call from her home in New York. Googling the song’s title, Holy Man on Malibu Bus Number Three, she discovered the mellifluous singer was Dory Previn: a Hollywood lyricist turned troubadour whose witty, incisive and deeply personal music ploughed its own furrow in the 1970s, amassing a cult following in spite of modest sales.

“She should be known, you know?” Greenberg says with a passion that she has funnelled into her documentary film, titled Dory Previn: On My Way to Where, which chronicles the artist’s extraordinary life in music. As a fan of Previn’s singer-songwriter contemporaries such as Joni Mitchell, Randy Newman and Carole King, I lament to Greenberg’s co-director, Dianna Dilworth, that I’m not as familiar with Previn as I should be. “That was me as well,” Dilworth smiles on a call from Switzerland. “We hear that from everyone.”

Which is where this film steps in, shedding new light on a woman whose writing in her mid-40s – about her abusive father, mental breakdowns and marital betrayal – went further and deeper than the work of any of her younger peers.

“Maybe I am bizarre to some people, but I’m not all that bizarre,” Previn deadpans at the beginning of Greenberg and Dilworth’s cerebral film. Born in New Jersey in 1925, Previn endured a troubled childhood, subjected to abuse by her father (an ex-soldier who had been gassed in the first world war trenches) whose paranoid episodes came to a head when he boarded up the family in their home and held them at gunpoint for several months.

Despite this trauma, or perhaps because of it, she defied stereotypes from the outset. At the age of 17, she got on a train to Hollywood and, after moonlighting as a chorus girl, landed a job as a lyricist at MGM in the late 1950s. It was there that she met her songwriting partner and soon-to-be-husband, André Previn. Earning 10 Oscars and acclaim as a pianist, composer and conductor, André became the better-known name, so this film aims to establish Dory in her own right.

Previn co-wrote songs for Tony Bennett and Judy Garland, as well as Doris Day – who became a mouthpiece for Previn’s frustrations, on the brink of feminism’s second wave, when she jauntily sang: “Control yourself, contain yourself, restrict yourself, restrain yourself.” A much-lauded soundtrack for the pill-popping melodrama Valley of the Dolls followed in 1967, a year that was between two nervous breakdowns for Previn: one in 1965 and the other in 1969 when she discovered that the actor Mia Farrow was pregnant with André’s child.

What came next would dispel not only the sexist cliche of the “spurned woman” but the stigma surrounding schizophrenia, with which she was diagnosed. While on a psychiatric ward, she was encouraged by doctors to write as a form of cognitive therapy. “She worked out her life through words,” Greenberg says – something that is brought movingly to life in the film through the use of diary entries and lyrics that pirouette across the screen. And yet, Dilworth adds, “It’s not as simple as ‘she wrote herself back to health’, which is a narrative that a lot of interviewers put to her. I think her writing led her to accept that she had many voices speaking in many different ways in her head.”

Or, to quote Previn herself: “When I was most unlucid and unclear there was always another eye, another part of me, that had absolute clarity.”

Previn’s relationship with the voices in her head – voiced in the film by J Smith-Cameron, AKA Gerri in Succession – developed through her music-making as she began to work with them, not against them, writing music to, in her own words, “continue living”. Her 1970 solo debut On My Way to Where, and her 1971 follow-up Mythical Kings and Iguanas – composed on a $60 guitar – best evoke this. Whether exploring fatherly abuse (“my daddy says I ain’t his child / ain’t that something / ain’t that wild”), rough sex (“one was an artist / one drove a truck / one would make love / the other would fuck”) or schizophrenic voices (“Mr Whisper’s here again”), she explored every facet of her life with craft and humour, and always with the great American songbook in mind.

“There is no artist that crossed over from Hollywood lyricist to singer-songwriter” like Previn, Greenberg says. Which makes you wonder: why wasn’t she more celebrated for it?

“It’s complicated,” Greenberg says. For starters, she was deeply afraid of travel and flying, which meant she rarely toured, and her overt sexuality and references to mental illness meant “men couldn’t deal with what she was giving us”. When her fourth record, Mary C Brown and the Hollywood Sign, was released in 1972, “a critic [for Creem] wrote that she flails a cliche with barbed wire,” Greenberg adds. “When Dory does it it’s flailing, when Dylan does it it’s genius.”

She went on to Emmy-winning songwriting in the 80s, then writing, lecturing and even a 1997 collaboration with her ex-husband, prior to her death aged 86 in 2012. Today we’re perhaps wiser to sexist double standards and sensitivities around mental health, making Previn’s story ripe for reappraisal. By facing down her demons with wit and honesty, she showed us alternative ways to live. As Previn said: “Who knows what the real reality is?”

• The nine-city UK cinema tour of On My Way to Where, as part of the Doc’n’Roll festival, begins 26 October, Barbican, London