There are additional connections with the TS and the Templars.
The disappearance of the shroud after the sack of Constantinople falls in the period of the height of the Templars.
The Ancient Templars were founded around 1118 A.D., and the last Grand Master, Jacques DeMolay, was burned at the stake, signaling the end of the activity of the Ancient Templars on March 18th 1314, after years of torture and trial. This corresponds roughly to one of the two periods when the Shroud was missing from the public view if you believe the Cloth of Edessa, the Mandylion, and the Shroud of Turin to be the same object. During this time also, coinciding with the establishment of the Ancient Templars, the idea of Chivalry as defining a code of conduct: moral, religious, and social became popular.
http://www.knightstemplar.org/KnightTem … ROUD.htm#9
The Holy Grail stories also arose during this time.
During this time period, another phenomenon occurred, the publication of what is known as the “Grail Stories.” These stories, including those of King Arthur and his round table, seem to link the concept of chivalry with a search for what is called the “Holy Grail.” The Grail, always associated with Christ, is at different times said to be the cup used by Jesus during the last supper, some sort of cup or bowl in which were caught drops of the Savior’s blood as he hung on the cross, or the actual bloodline of Christ who is said to have fathered a child by his wife, Mary Magdalene. No matter which legend is proposed, it always has reference to something that contains the blood of Christ.
http://www.knightstemplar.org/KnightTem … ROUD.htm#9
The association of the Holy Grail with the Templars has precedents even in 12th-century fiction; Wolfram von Eschenbach’s Parzival calls the knights guarding the Grail Kingdom templeisen, apparently a conscious fictionalisation of the templarii.
https://en.wikipedia.org/wiki/Knights_Templar
Trials of the Templars referred to a worship of a “head”.
The Templars had their own internal clergy, and their modes of worship may well have developed in ways somewhat independent of the main stream church. They also had initiation ceremonies known only to them and were questioned extensively about these ceremonies during their inquisition and suppression. In our series about the trials of the Ancient Templars, we published a synopsis of the transcript of these trials, and one of the questions put to these Templars concerned the worship of a “head” or an idol in the form of a head. This concept is still used in anti-Masonic propaganda where the head is now given the name “Bophamet” and drawn as the head of a goat! Modern Freemasons are thoroughly confused by such accusations and completely mystified as to where something so bizarre may have originated.
http://www.knightstemplar.org/KnightTem … ROUD.htm#9
Initiation of Templars involved “denying the crucifix”.
Almost all of them indicated in testimony that they had been told to “deny this” referring to a crucifix held in the initiating Knight’s hand. This worried me as all other evidence indicated that these ancient Templars were among the most devout Christians of their time. Why would they tell new initiates to “deny” a symbol of the Savior’s sacrifice for their sins? I now believe that this rite was to point out to the candidate for Templary that on the Templar’s altars, an image of the Shroud, the symbol of the resurrection of Christ, had been substituted for the crucifix, an emblem of the death of Christ. It was as if to say to the young Templar that anyone can die, but only Jesus, the Savior, can resurrect again to life. This would impress on the initiate that it was the resurrection of Christ, not the crucifixion that distinguishes Him as the One who has power over life and death and Him alone through whom salvation is obtained. The Shroud was used as a symbol of the resurrection, replacing the crucifix which was a symbol of His death.
http://www.knightstemplar.org/KnightTem … ROUD.htm#9
The Templars did many things secretly. And if they possessed the shroud, they would not have revealed it to the outside world.
Above all, the Templars held all of their deliberations and votes in secrecy.
https://www.franciscanmedia.org/st-anth … s-templar/
Eventually, their rules of secrecy, their power, privileges and their wealth, made them vulnerable
https://en.wikipedia.org/wiki/Trials_of … ts_Templar
https://debatingchristianity.com/forum/viewtopic.php?p=1122564#p1122564