The result is generally magnificent, if you want it to drive like an old car but better. Icon does brisk business with those folks. Some, though, really just want a new vehicle that looks old.

Volvo’s first sports car was a flop in its day, but the quirky P1900 has since become one of the rarest Volvos ever built.

Carla Kidby unrolls a foreboding black mat of plastic spikes. “This came this morning,” she says, brandishing her spurs as if ...