Can't see them using a costly piece of linen to cover a bloody body on a stretcher. I suppose it's conceivable, but wouldn't it have been kept and mentioned in the Bible? For that matter, if an image appeared on anything touching Jesus after the crucifixion, it would certainly have been mentioned in the Bible. It seems to me that somebody took the story about the Veronica and decided to make a full body image on a shroud similar to the image described in regard to the Veronica. The Shroud and Sudarium were produced at the same time, since they match, and both first appear in history shortly after 500 AD. Oddly, they didn't put the face image on the Sudarium too. That shows that Jesus didn't really leave his image on any cloth that touched him.
It's possible that the Shroud and Sudarium really did cover the body of Jesus, but that the Shroud image was added later, to match up with the already present blood stains. This may explain why there is no image layer beneath the blood spots, as reported by scientists.
The person who had those cloths on him was definitely dead, as mentioned in these facts about the Sudarium;
It seems to be a funeral cloth that was probably placed over the head of the corpse of an adult male of normal constitution. The man whose face the Sudarium covered had a beard, moustache and long hair, tied up at the nape of his neck into a ponytail.
The man was dead. The mechanism that formed the stains is incompatible with any kind of breathing movement.
The man was wounded before death with something that made his scalp bleed and produced wounds on his neck, shoulders and upper part of the back.
The man suffered a pulmonary edema as a consequence of the terminal process. The main stains are one part blood and six parts fluid from the pulmonary fluid.
The only position compatible with the formation of the stains on the Oviedo cloth is both arms outstretched above the head and the feet in such a position as to make breathing very difficult, i.e. a position totally compatible with crucifixion. We can say that the man was wounded first (blood on the head, shoulders and back) and then 'crucified.'
On reaching the destination, the body was placed face up and for unknown reasons, the cloth was taken off the head.
The Sudarium contains pollen grains of Gundelia tournefortii, identical to that found of the Shroud that grows only east of the Mediterranean Sea as far north as Lebanon and as far south as Jerusalem.
The blood (stain symmetry, type and other indicators) on the Sudarium matches the blood on the Shroud. http://www.shroudstory.com/faq-sudarium.htm