Most people consider the Fall a post-punk band, but its members always seemed to act like a pop band that wasn’t exactly popular. Ever since forming in 1977, the Manchester band and its lone constant presence, Mark E. Smith, have carved a topsy-turvy line through popular music, a comet streak especially visible and relevant in England. Despite an extensive discography and the official live concerts, bootlegs, and reissues that the CD age has added to that output, a portion of the Fall’s elusive appeal has always escaped American ears.

“The Complete Peel Sessions 1978-2004” provides a peek into the radio-pop workings of a band that over the years has gone through more facelifts than Michael Jackson. The Fall appeared on John Peel’s BBC show 24 times, the first just a year into the band’s live existence – exposure that put it on the British music map – and the last but two months before Peel’s untimely passing.

The six CDs of this box set offer snapshots of every one of the band’s convulsive personality tics – the lineup changes, the stylistic experiments both profound and ill advised, the flirtations with traditional pop, the outrageous lyrics, the interpersonal dramas, the quasi-political outrage, and the outright venom. But the biggest surprise here is the mirthful jubilation with which Smith and the band execute their anti-pop rebellion.

Like the novels of Martin Amis or the movies of Mike Leigh, the music of the Fall is decidedly British in ways that can escape outsiders looking in. The May 2, 1994, Peel session featured a version of the band recently freed from major-label recording and moving away from pop. The return of a harsher sound rekindled Smith’s scalpel-sharp tongue: “M5,” named after the British highway that runs from London to the countryside, is a bopping if vituperative mash note to living in the city. “This is not an autobahn, it’s an evil roundabout,” Smith speaksings in his clipped tongue. “That leads to the Haywain / and you’ll never see good trains again.”

Even better from this session is a reworking of the Fall’s early song “Hey Fascist” into “Hey! Student” (off 1994’s “Middle Class Revolt”), an excoriating diatribe against middle-class hipster youth. Mainlining the propulsive, punky throb of “Hey Fascist,” the song twists the original spite against those with “Car coat on, steel boots on [their] feet / Write you letters to the Evening News” to the feckless youth with “long hair down and sneakers on [their] feet.” In this Peel session, Smith, in a moment of ribald self-editing, twists the lyric from the studio version, “as you listen to Pearl Jam in your room” to “As you listen to Pearl Jam in your videodrome / I feel my knife and sing this song.”

That’s not exactly belly-laugh material, but Smith’s humor leans toward the dry. Listen to the band’s two holiday covers for the December 12, 1994, session, where it just murders “Jingle Bell Rock” and “Hark the Herald Angels Sing.” The former becomes a choked Ramones song that speeds along in just over a minute before ending like a car crashing into a tree. The latter may be the funniest thing the Fall ever committed to tape.

Smith has never aimed for the musical with his voice – he favors regular, familiar pop-song meters, but sings them mushy-mouthed and masticates short vowels into grunting belches – but he really goes for the unappealing on “Hark the Herald Angels Sing.” It’s hard to deny his success: He stammers through the carol over the band’s jaunty, mid-tempo version of its melody, and the former Mrs. Smith, guitarist Brix, massacres the titular chorus in a bleating majesty that sounds like a partridge being flattened by a pear tree.

The band had ups and downs in the ensuing decade, but four songs from an August 2004 Peel session feature a reinvigorated Fall firing off noisy, uptempo darts that recall the band’s earliest triumphs. “What About Us” and “Blindness” are as strong as anything from the band’s fertile early 1980s era, as is a cheeky cover of “I Can Hear the Grass Grow,” a song by the late-1960s British outfit the Move. The strained harmonizing on the chorus is one of those rare occasions where the Fall is actually trying to make somebody laugh – and wildly succeeds.

The final session is a confounding, daft distillation of the British group’s peculiar gifts. Recorded for Peel’s 65th birthday celebration, aired August 31, 2004, the dub-like, meandering drone of “Job Search” plods along, propelled by an electronic bass line that dances awkwardly with a low electronic hum reminiscent of a sick synthesizer. Ghosts of strings haunt a guitar strum that plinks through the background as if in a different song entirely.

The lone thing holding the song together is Smith’s obtuse lines, which he sings as if chewing a tough bit of steak. His haggard voice mashes syllables and camouflages enunciation, leaving only snippets of the lyrics intact. Like a slide version of a whole movie, the song becomes a collage pieced together by the brain. It would be a total disaster if the Fall – and fans of its undoubtedly acquired tastes – didn’t take such delight in denying the pop song’s usual pleasures.

These moments of childlike and admittedly childish mirth add a new wrinkle to the Fall’s profligate output, adding a sense of humor to the tremendously stark and occasionally dour recordings of its first decade. Happily, though, the first 10 Peel sessions provide the monolithic Fall songs of legend – “Hip Priest,” “Eat Yrself Fitter,” “New Face in Hell,” “C.R.E.E.P.” – with their spite intact. And the ramshackle, spirited jolliness of the Fall’s undisputed classics from its June 15, 1978, debut session – “Industrial Estate” and “Rebellious Jukebox” – leave as indelible a mark as its sharp-tongued, blistering intelligence. No wonder Peel invited them to the studios so often.