Look at the carbon dating tests. They took a few threads from it and dated them to exactly the century when the Shroud is first historically documented. With any other piece of cloth that would be it. But people who desperately wanted it to be genuine went the extra mile and figured out how it could have accidently picked up some extra carbon-14 during a fire. Now there is research going on about the effects of smoke on Carbon-14 content. Would anyone bother with this if the authenticity of the Shroud of Turin weren't involved?
So rather than being something special about the Shroud itself it's simply the quixotic nature of human knowledge that is being demonstrated. And I think that is what the 'trickster' archetype is really about. The Shroud is just the accidental focal point.