I wonder about this because a few years ago on a one-country-a-day European cruise I chose a day tour of Monserrat. There's a definite spiritual feeling there, and some of the relics and construction go way back.
If the theory is correct that what we now know as the Shroud was previously known as the "Image of Edessa" and the "Holy Mandylion", then it s fairly well documented.
The image would have been brought to Edessa in the years after the Resurrection, then brought to Constantinople, then stolen by Crusaders during the sack of that city in 1204, after which it was hidden until the mid-1300s.
I think most of the favored theories involve the identity of the Shroud with the Mandylion of Edessa, which was in Edessa in modern Turkey and then Constantinople for the first millenium AD, only being brought to the West by the Crusaders after the sack of Constantinople. I don’t know what legend says of the travels of the Sudarium of Orvieto.