Synchronicity strikes again! A friend in the US (not a Fortean) emailed me about the program that Iopaka saw. As it happened, I had just finished reading "The Second Messiah" (A Google search will turn up reviews, etc.) which covers the shroud, Templars, Freemasons, Rosslyn Chapel, Arthurian lore, etc, etc.
And this is the second thread today I've had cause to mention this book, which I highly recommend.
The basic premise is that the Templars derived from families descended from survivors of the Jerusalem Church (which did not believe Jesus was divine, unlike most of Christianity today, which derives from the teachings of Paul, who never even met Jesus).
When the French King swooped on the Templars, their leader was cruelly tortured (including flogging and crucifixion), but then left to recover wrapped in a shroud in a grotesque imitation of a Templar ritual. He eventually recovered, the shroud was washed and put away. Generations later it was discovered that his image was imprinted on the shroud, and the owner put it on show to earn some money. (This post greatly condenses the detailed evidence the authors amassed.)
So the shroud is medieval, in accordance with the carbon-dating, according to this book. Regular Christians won't like it because it denies Jesus' divinity, and disconnects him from the shroud. But to little ol' agnostic me (who has read widely on the subjects mentioned here) this book stitches together the most consistent story I've ever read out of all the little fragments of historical evidence we have left.
The other thread I mentioned this on is about the Oak Island Money Pit, where (some say) the Templar treasure ended up.