In 2003, Chad Friedrichs’s documentary, “Jandek on Corwood,” examined the world and discography of Texas’s weirdest, most mysterious musical son (and when one considers the Legendary Stardust Cowboy or Ornette Coleman, there have been plenty of odd ones). Little was known about the man called Jandek, but there was plenty to hear.

Starting with 1978’s “Ready for the House,” no fewer than 33 albums had been released to that point (15 more have landed since) on the Corwood imprint, on which Jandek is the only artist. Each featured eerie ramblings that sound at best like something made in a garage (perhaps by Leatherface), and blurry Kodachrome images (a shirtless man with giant sideburns, an unkempt back yard, a drum kit at dusk) that only deepened the mystery.

At the time of Mr. Friedrich’s film, Jandek had never given an interview or played a note in front of an audience. He didn’t even appear in the film, though a “representative of Corwood Industries” cooperated with the production — though no one would know if it was Jandek to look at him. Such a mystery in an age of media saturation led many obtuse critics to laud the man, whose real name they didn’t even know.

It helped that the very task of listening to his music made most people cover their ears, so that being a fan was itself a badge of honor. It was one thing to listen to one side of a deep but tuneless baritone screaming “Don’t paint your teeth!” backed by someone who had never hit a drum in his life — but an ever-burgeoning discography stretching into the 40s?

“Detuned,” “deranged,” and “psychopathic” are common words flung at Jandek’s body of work, and yet as alien and odious as his albums may sound to initiates, the world has beaten a path to his door (wherever that is) in many ways. There are moments, on an album like 1986’s “Telegraph Melts” that could get mistaken for Sun Ra or a Jefferson Airplane bootleg (poorly recorded, of course). On albums like 1989’s “The Living End,” one can hear in Jandek’s lo-fi ramblings and highly personal songcraft the forebear of artists from Lou Barlow to Cat Power.

Such a persistence of vision now informs the New Weird America scene, with folks like Sunburned Hand of the Man and Wooden Wand churning out spare, atonal albums as they see fit. Jandek’s peculiar tunings (or nontunings) and rhythms evoke the era of Texas bluesmen like Lightnin’ Hopkins, who played in keys and rhythms entirely of their own device. And when it comes to his a cappella albums (see 2000’s “Put My Dream on This Planet”), he’s still entirely in his own world.

By the time of the documentary’s release, 25 years after Jandek released his first album, the mystery of his identity was completely intact. The film’s talking heads, from All Music Guide’s Richie Unterberger to music scribe Byron Coley, took cold comfort in that any real information about the subject might never come to light.

And then a funny thing happened. At an arts festival in Glasgow, Scotland, in 2004, that “representative from Corwood” appeared onstage, looking remarkably like the eerie, gaunt man that figured on nearly every album cover. Playing with a clutch of local musicians, he both looked and sounded like Jandek — which is to say it couldn’t sound like anyone else. The man did not address the audience, and he didn’t play any “hits.” Instead, he presented an entirely new batch of inscrutable, harrowing songs.

It was not an isolated event, as this “representative” began to appear much like a UFO, recruiting musicians (generally of the free-improv idiom) who would scrabble to keep up with the man. He appeared in Seattle, Chicago, and Austin, Texas, each time recruiting local players. With every concert appearance, an album followed. His debut performance became “Glasgow Sunday” (album no. 41), while a return engagement in Glasgow (to present an eloquent 90-minute suite for piano, bowed bass, and percussion) became of course, “Glasgow Monday” (album no. 47).

Jandek played New York previously (with No-Neck Blues Band bassist Matthew Heyner and free jazz drummer Chris Corsano), but his concert this Saturday at the Abrons Arts Center (an off-shoot of the downtown music club Tonic) with Dirty Three drummer Jim White and Two Dollar Guitar’s Tim Foljahn should pack plenty of surprises. On a weekend when another lost Texan, the 13th Floor Elevators’ Roky Erickson, makes two New York appearances, one can perform the rare trick of seeing two vaporous Texas legends walk among us. You may never get the chance again.

Jandek will perform Saturday at the Abrons Arts Center (466 Grand St. at Pitt Street, 212-598-0400).