Will Karlheinz Stockhausen’s reputation ever be rehabilitated? The question hangs over this city this month, with the world premiere of “Sonntag aus Licht,” the last, longest entry in the composer’s 29-hour, seven-opera cycle, “Licht” (“Light: The Seven Days of the Week”). And though the German composer — a leading figure in contemporary music for the second half of the 20th century — died in 2007, for some, particularly in the United States, he still lives in the 9/11 penalty box.

In late September 2001, after a journalist questioned him about the relevance of characters in his mythological operas, Stockhausen responded with an ill-advised metaphor. Speaking about his Lucifer — a nasty-if-ingenious force for destruction in the “Licht” saga who “does not know love” — Stockhausen posited that the character’s handiwork could be observed in New York and went on to call the attack “the biggest work of art there has ever been. . . . Just imagine what happened there. There are people who are so concentrated on this single performance, and then five thousand people are driven to Resurrection. In one moment. I couldn’t do that.” Stockhausen started contextualizing quickly, by reminding his audience: “It is a crime, you know of course, because the people did not agree to it. They did not come to the ‘concert.’ That is obvious.”

But his meaning was not obvious to everyone. The shortened quote — “the biggest work of art there has ever been” — traveled the world quickly, in some instances minus anything about Lucifer being “the artist” in question. Concerts were canceled, and Stockhausen was forced to clarify that he did not support terrorism. But the composer had shown a hint of what the critic Richard Taruskin had diagnosed in a 1993 essay for the New Republic as a “hypertrophied aestheticism that transgressed the normal boundaries of art and invaded life.” Today, if you utter Stockhausen’s name in polite American classical company, you can still reap a chorus of “You can’t be serious!”

And so it fell to the composer’s hometown crowd to rally to the cause of bringing the “Licht” cycle one step closer to completion. By the time of his death in 2007, the composer had seen five of these works staged. The last world premiere of a entire “Licht” opera — each of the seven is named for a day of the week — took place in 1996; since 2001, rumors of complete performances to be mounted have been snuffed out almost as soon as they were first whispered.

All that changed last weekend in Cologne. Before the show began, each member of the audience was invited to choose a canvas-backed reclining chair placed evenly around a cylinder in the circular, stark-white auditorium. Up next to the ceiling, six white screens resembling fan blades were attached to that same cylinder, which also anchored a movable metal platform that held video projection equipment and was operated by men wearing spacesuit-like gear and vaguely astrological-seeming facepaint.

Once members of the orchestra had filed into the auditorium, a woman, the character Eve, appeared. Or rather, six women, walking as one inside a pyramid-like latex suit that allowed stray hands and Kabuki-painted heads to become part of the costume’s body. At the top of the massive macro-woman was the soprano Anna Palimina. Across the room from her was Michael, portrayed by tenor Hubert Mayer, who was dressed even more improbably. He stood inside a contraption that looked like a Segway, but which quickly — and somewhat phallically — hoisted him on a hydraulic spit, near the top of the auditorium’s domed ceiling, before twisting him at a 45-degree slant, where he was held aloft.

Then the video projectors switched on, revealing animations, along the walls and on the ceiling, of planets and moons from our solar system. Stockhausen was back — hoping to transport us to another realm. It was time, finally, to release him from the penalty box and let him have his shot.

“Sonntag” (“Sunday”) is so long — more than eight hours, with intermissions — that for most of the performances in Cologne this month, it is being presented over the course of two consecutive evenings. What can be seen and heard there is as multifaceted as the man himself: often breathtaking in its musical innovation, while at other times almost insultingly vapid on the intellectual score.

“Sonntag aus Licht” (“Sunday from Light”) mostly works as a recapitulation of the other “Licht ” operas’ themes, with snatches of melodies associated with characters and their stories mingling and bumping up against one another. Like Wagner, Stockhausen has some half-hours that drag on and on. But unlike in the “Ring” cycle, when some useful information is being communicated (slowly) about a sword or a metal hat, in “Sonntag” the slow moments are just vague exclamations of praise to God and the universe.

It goes on like this: in Arabic, in Chinese, in English, etc. The music is, at times, hyponotically great. Particularly during the second scene, when an a cappella choir moves through baroque fillips and Gregorian-style drone while managing 20th-century polyrhythmic complexity.

The one missing character in this scene, and in most of “Sonntag,” is Lucifer. It’s tempting to wonder whether Stockhausen, who worked on “Sonntag” until 2003, banished the character from his final work out of sympathy for — or at least acknowledgement of — his critics. Lucifer appears most directly in Scene IV, “Scents-Signs,” in which seven scents are wafted out at the audience (disappointingly, most smell like generic incense or flavored hookah smoke). Finally, he is encircled within a ring of fire, trapped. This devil is not getting out to do any more destruction, the production seems eager to assure us.

During the premiere weekend, the strongest scene proved to be the opera’s climax, “High Times for Chorus and Orchestra.” (The word “Hochzeiten” in German means “weddings” but can also be read as “high times.”) And yet the scene is actually two connected scenes, since it is played twice by five small chorus groups and five small orchestral ensembles (all playing in different tempi). The choral and orchestral forces are segregated into different auditoriums. And in each hall, seven “inserts” — or weddings — of music from the neighboring auditorium are achieved via a live relay, piped in through loudspeakers. The audience is divided between the two forces, then during intermission changes halls to hear the music from the other sonic vantage.

Unlike some of the more drearily cosmic passages of “Licht,” “High Times” serves a recognizable spiritual purpose. Presented only a little over a mile away from Cologne’s architectural marvel of a cathedral, “High Times” better manages to evoke Saint Augustine’s famous command to “hear the other side” than any of the reliquaries in the Gothic church that took over half a millennium to build, just across the Rhine.

It all speaks to why America ought to consider Stockhausen anew. Already, a younger generation of classical composers and musicians have been discovering the composer on their own terms. The Respect Sextet, an improvising group of conservatory-trained players, had the sharp idea to record pieces by Stockhausen and Sun Ra — two men seeking “other planes of there” — alongside each other. The ensemble Alarm Will Sound has produced an evening-length work, “1969,” which imagines the “what if?” of a rumored but never realized meeting between John Lennon and Stockhausen. In addition to a version of Stockhausen’s “Hymnen,” the show contains an acoustically orchestrated version of “Revolution No. 9” from the “White Album.” The man behind that arrangement, the young New York composer Matt Marks, also has a new amplified string quartet coming in May, which uses recorded snippets of ’80s pop songs in a way that recalls, in form if not tonal approach, Stockhausen’s use of national anthems in the electro-acoustic “Hymnen.”

This is why Stockhausen continues to matter in a cosmic way for music, if not in the actual cosmic scheme of the universe that he wanted to overpower via his aesthetics. You can’t get the shmear of strings that bridges the A and B sections of “A Day in the Life” without his piece “Gruppen,” written for three orchestras playing at once. (That’s why the Beatles put him on the cover of “Sgt. Pepper’s.”) And in his autobiography, Miles Davis credited Stockhausen’s compositional philosophies with the radical turn he took circa “On the Corner.” In the decade since his 9/11 moment, there’s no telling what we’ve missed out on — not just from the catalogue of late-Stockhausen music that has yet to be performed stateside, but the music that it, in turn, might inspire from other artists.

The good news is that there’s still one Stockhausen “Licht” opera yet to have a world premiere. That would be “Mittwoch aus Licht” (”Wednesday”), in which members of a string quartet take off in different helicopters from the stage while playing in synch. Santa Fe Opera, given your outdoor auditorium and all, your mission could hardly be clearer.

Walls is a freelance writer.