In 1989 some Hollywood producers rang up Howe Gelb, founder and leader of the Arizona band Giant Sand, and told him he had to come to an empty theme park north of LA immediately to teach Keanu Reeves and Alex Winter guitar parts for the movie Bill & Ted’s Excellent Adventure. It was a fantastic opportunity, they told him, that he couldn’t afford to miss. ”Yeah,” remembers Howe, ”but my kid Indiosa had a fever that night, and it was so much fun to say ‘no’.”

This might go some way to explain why you’ve never heard of Giant Sand, and, conversely, why the woman at the check-in desk of the Hotel Congress, 311 East Congress, Tucson, speaks of Gelb with the kind of pride one would normally reserve for a local area of outstanding beauty.

Formed in 1981, Giant Sand were minor players in a sudden wave of great American guitar bands, of which only REM became a household name. At the same time, Gelb was also releasing quirky and then suicidally unfashionable country and western records under the absurd pseudonym of Blacky Ranchette. In the late 1980s Howe put Blacky out to grass, married his two alter egos together, and signed up John Convertino, the apparently telepathic drummer who lived downstairs, to give Giant Sand a sound that is entirely its own electric-acoustic, semi-improvised, multi-rhythmical country rock.

At best, the Giant Sand sound is eclectic to the point of being thoroughly uncategorisable. Former members of Howe’s band have included pop-punk Evan Dando of the Lemonheads, country singer Lucinda Williams, who has written hits for Mary Chapin Carpenter and Emmylou Harris, and an ageing crooner called Pappy Allen. “I moved to a little town called Rimrock, just a smattering of houses in the desert,” recalls Howe. “Pappy was the first person I met there. He was 74 years old and we ended up taking him off round the world.”

The 1994 album, Glum, includes a bewitching cameo from Howe’s daughter, Indiosa, called The Bird Song. It’s a whole new music Infant Free Jazz. “I kept dogging her to sing. She was five at the time. I used to play guitar to her to get her to sleep. One night she sang seven songs into a tiny recorder, `the bat song’, `the monkey song’, `the tree song’, while I tried to keep up with her. Consequently you’ve got an album with a five-year-old and a 74-year-old, both putting us to shame.”

Live, Giant Sand have the same unpredictability. Their 1993 tour, with a massive eight-piece line-up, was an unbelievably classic rock show, while a three-piece gig in London in 1994, in which they attempted to duplicate their largely improvised Purge & Slouch album, saw them get so lost they had to leave the stage to regroup. At their worst, Giant Sand are like some punky teenage daughter who is a secret violin prodigy, but whenever curious relatives come round she sulks in her room and refuses to perform. So is Gelb working to a plan?

”Hmmm. I met Peter Buck from REM in Seattle last year. He asked me what my ambition was, and I realised I didn’t understand the word.

I understand words like desire, hunches, instincts, but not ‘ambition’. It sounded like a good word. I realised if I didn’t use any ambition, then I was just going to stay bent on that whole self-destructive thing. It was in my nature to do that. But things are changing. Last year I fell in love with this girl, but I denied it, and chased her away. I was 39. You don’t think you’ll fall in love then, just like you aren’t going to get a tattoo on your face at 40. But it made me realise I was doing myself in. I thought about all the people who value the band and how half the time I was disregarding them, just driving us off into a ditch.”

It appears that Gelb has a safety valve after all. He even took the Keanu Reeves job in the end, after they arranged the transport: ”Keanu was a great intuitive bass player, like a young Jack Bruce, but Alex Winter just tried to duplicate my stuff note for note. Hey, I can’t even play my stuff note for note.”

Barbecue, Giant Sand’s new album, culled from live radio sessions, features dainty, folksy melodies, slouching psychedelic country blues and swathes of strange steel-guitar-led lounge music, all stretched to breaking point. Though an accurate document of the stripped-down live Giant Sand at their most inventive, it probably isn’t a great place for new listeners to pick up the plot. So which is Howe’s best record? “Probably Glum, from 1994. It amazes me how well it fits in with all the crap I was going through with this girl, even though it was recorded before it happened. Do you think we can have memories of the future?” I don’t know, I answer. Either way it isn’t something we can deal with in 800 words. “It’s as if I could scent the fire, smell the smoke of wood that will be burning in the future.”

Barbecue is released on Normal records; Giant Sand are at the Mean Fiddler, Harlesden, London, on Thursday.