It’s a moment to cherish, a moment to remember: Richard O’Brien is singing to me. He’s laid himself bare in ways that don’t always have obvious connections with The Stripper – his latest musical project, which opens at The King’s, Glasgow, on September 29 – and he’s been valiant and dutiful in reconnecting with The Rocky Horror Show which will, in yet another touring production, play Glasgow later in the year.

But as our time together draws to a close, the witty wordsmith lyricist in O’Brien comes to the fore. He’s spoken, on and off, about The Stripper with relish and affection – described this musical foray into the realm of pulp fiction in terms of “blondes, bullets, burlesque” albeit with a jazz-inflected score that would, he reckons, have been a choice vehicle for Frank Sinatra back in the day … Say 1961, the year The Stripper is set in, and suddenly O’Brien is himself crooning. Laying out the final number in the show, with its wistful theatrical flourish in the closing lines – “Now I know just what I’ve lost and I know just what it cost … ’Cos that’s the way the play came down.”

There’s a pause for thoughts – emotions too, probably – to be gathered together before a soft-voiced O’Brien murmurs: “That’s a crime line, isn’t it? ‘The way the play comes down...’ A kind of pulp fiction phrase. But it’s also part of this theatrical metaphor that runs through the songs. And I rather like it. For instance, when Dolores – our lead blonde stripper – dies, there’s this song I’m Planning My Final Exit. It begins” – and without missing a beat he’s singing again – “‘You come on with a smile and a song/To find out that it’s all going wrong …’”

And it could just as easily be a verse and a chorus about his own life and career, about the rollercoaster highs of stage and screen success that couldn’t compensate for the lows when, in his own assessment, he “lost the plot and stepped over the edge of the abyss. I descended into paranoid delusion. This was five or six years ago. But really, I was always torn apart. Always battling, trying to understand why I am the way I am.”

Now, the rest of us might rush to fill in all kinds of adjectives here. Assume that the man behind the Rocky Horror phenomenon, feted globally for both his writing and his incarnation of Riff-Raff – was talking about creative demons or crises of the “writer’s block” variety. But no. O’Brien is unstoppering the pent-up misery of decades, clearing the air – and his mind – of the unhappy issues that attend someone who is “trans”. He means transgender. And it’s this state of disrupted being, and how – at the age of 67 – he is now dealing with it, that punctuates almost every aspect of our conversation about The Stripper.

You could say it’s because in life as in art – even as in escapist musicals and the dime novels churned out by Carter Brown, author of The Stripper – appearances are everything. Brown’s crime stories, featuring strippers, cops and sexy secretaries, matched his readers’ expectations of life on the wild and seedy side of 1950s America. He wrote more than 150 of them with sales topping 70 million copies – and if he was more popular in Europe than America, well someone else’s “mean streets” are always grittier and more romantically noir than one’s own backyard lot of lumbering thugs with menacing intentions.

When it comes to O’Brien’s own appearance … well, on the morning we meet, he’s a dapper conundrum: undeniably stylish in tobacco-tweed jacket, skinny white jeans – tucked into some expensively tooled cowboy boots – with a creamy silk shirt open at the throat so as a mass of heavy crucifixes can catch the eye. “If I was a woman wearing this, guys in the street would want to pull me,” he says in a dryly matter-of-fact of voice. “Instead, I’ve had grown men sniggering at me. They don’t know me. They’ve got no reason to be so offensive – and it does offend me. Because this is who I am.

“All my life, I’ve been fighting, torn in two and battling – never belonging, actually. Never being male. Or female. Wondering if I was born transgender? Did it happen in the womb? That might have made it easier. I don’t know. Or was it psychological? I’d been going to therapy, treating what I was as though it was some kind of illness – getting more and more depressed, wondering, ‘could I be cured?’ I went mad, really. My marriage was going down the tubes and I just lost it. Lost it. Lost it...”

If this were some Broadway musical or a Hollywood tear-jerker, this might be the moment for O’Brien to cry “hallelujah!” and attribute his current state of improved wellbeing to a religious conversion, a new relationship or maybe even landing a key role in some crowd-pleasing blockbuster. Instead he credits his toe-hold on happiness to a combination of his own – and his children’s – acceptance of his “in-between-ness” – and courses of oestrogen. “I think the oestrogen has made me an altogether nicer person,” he laughs, adding: “and it’s made my skin wonderfully soft. I was always blessed with good skin, but now it’s had a real boost. Feel.” I stroke his cheek, alabaster smooth and as pale and wrinkle-free as the shaven head he made a cult look long before Kojak became a popular TV personality.

He feels sorrow and exasperation at the way society imposes a sense of “being abnormal or perverted on those of us who are, as I like to think of myself, a ‘third sex’. Once I’d settled on that concept, I found it freed me from a lot of inner conflicts. But what a shame, that you have to hide yourself away. Become a shadow, so as you don’t make any waves. I don’t see any point in making yourself invisible – I’m too vain! And yes, from time to time, I put a frock on and make myself look gorgeous – exceptionally gorgeous – and my daughter, she’s 20, will be with me and when anyone sniggers or points, she says, ‘That’s my dad. It’s the way he is –right?’ As long as my children love me – and they do – I’m fine. And I’m happy.”

Luckily for his fans, being on stage can make O’Brien happy too. The vanity he preens himself on apparently gets discarded in the wings. “Ohhh – I do love to entertain,” he says, and the voice suddenly throbs with a gravelly honeyed timbre. “But I don’t take ego on stage with me. It’s not about showing off, it’s about sharing. To be responsible for people laughing, having a good time, is truly wonderful. It’s not about feeding the ego. Though it’s probably about feeding something...” Silence. A wry smile. A slight shrug of eloquent shoulders.

His being on stage in The Stripper was, however, unintentional. After he and composer Richard Hartley had handed over the cast list, O’Brien realised he was a character short. So now he’s Mr Arkwright of the Arkwright Happiness Club, where lonely hearts seem destined to meet with an Awful End instead of Mr Right. When Detective Al Wheeler starts to probe the not-so-accidental death of a wannabe actress, he not only encounters the delectable stripper Deadpan Dolores, but sets in motion a series of events that ends with five people dead and Wheeler singing a closing number with a poignant, dying fall.

“It’s a rather brave, not to say risky, way to end a musical comedy, isn’t it?” says O’Brien. “We can always put in a breezy reprise for the curtain call – we’ll see how it goes. But really, I’m hoping audiences will just sit back, enjoy it all and think that it’s cool. Maybe sometimes a bit naughty, but naughty-cheeky, not offensive. It’s one of those 1960s smoky, sexy detective thrillers – the book was, I suppose, the paperback equivalent of the B-movies that inspired Rocky Horror. Only here, instead of rock’n’roll we’ve gone a bit jazz-swing/swing-jazz with the music, and the humour is a little wry’n’dry – a cool cocktail.”

Wry and dry. A cool cocktail. Richard O’Brien could, unwittingly, be describing himself. Look out for a vintage O’Brien cameo – including a brief frolic in a frock, apparently – when The Stripper comes to town.

The Stripper is at The King’s Theatre, Glasgow, from September 29-October 3.