What on earth would an innocent make of Richard O’Brien’s goofy, vulgar musical, a Halloween-season staple that earned cult status on the midnight-movie circuit? A spoof of old horror and sci-fi movies with a score that parodies ’50s-era music, the lewd, lighthearted comedy could be pretty offensive — if you didn’t know what you were in for.

But nobody at Stray Dog seems confused. Under the direction of Justin Been, Stray Dog presents O’Brien’s original stage show, which debuted in 1973. (The wildly successful movie version, released two years later, stars Tim Curry, Susan Sarandon, Barry Bostwick and Meat Loaf.) Been no doubt knew from the start that the audience would be filled with people who shout out lines and catchphrases, sing along with the songs and dress up in costumes like members of the cast.

They have a wonderful time. Even the Rocky “Virgins” seemed to know what was up; if nothing else, they must have guessed before the show, when members of the cast wandered around, lipsticking the letter “V” on their foreheads.

The story’s pretty simple. Two newly engaged naifs, Brad (Kevin O’Brien) and Janet (Heather Matthews), hunting for a phone when their car breaks down on a rainy night, stumble into the castle of Dr. Frank ’N’ Furter (Michael Juncal), an arcade of sexual abandon and weird “scientific” experiments. They make some amazing ... discoveries.

Juncal — his shaven head looking sort of canine over a pearl choker and his meaty legs encased in fishnet hose — portrays a more downscale “Sweet Transvestite” than Curry did. But it works for this production. Improvising lines and bits, wobbling down the onstage staircase in “do-me” shoes and seducing one and all, Juncal gives the play its far-off-center center.

The whole cast has fun with him, with particularly impressive work from Maria Bartolotta and Corey Fraine as the decadent servants Magenta and Riff Raff. Bartolotta sings so beautifully that she makes the opening number, “Science Fiction,” sound like a real song instead of a parody. But, loud and crude as Magenta is, Bartolotta manages to communicate an unexpected tenderness. (It would be great to see her tackle Serafina in “The Rose Tattoo.”)

Choreographer Zachary Stefaniak Shaffner gets the whole cast moving in style, especially in the splashy numbers “The Time Warp” and “Hot Patootie.” Costume designer Eileen Engel dives straight for the cellar in a giddy celebration of sexy sleaze. And Chris Petersen’s instrumental quartet, squeezed into a center-stage nook between a pair of staircases, handles O’Brien’s many musical modes with aplomb.

With the new Fox production on TV and midnight showings at the Tivoli in the Loop, a live production might be just what you need for a true Rocky-fest. But if you want to catch it, act fast. Few tickets remain.