What was the significance of Jesus folding the burial cloth?
Why did Jesus fold the burial cloth (aka napkin) that was around His head after His resurrection?
And a follow up question, why was it separate from the other cloths?
The easier question is why the cloth around His head was separate from the other cloths. I believe this has to do with convenience and simplicity. It is very difficult to wrap a dead body in anything. Trying to do so with a single piece of cloth would make it even more difficult. A long piece of cloth would have to be found, and the body would have to be moved and lifted more to shuttle the cloth under it for another turn. It is easier to find shorter pieces of cloth given the manufacturing technologies that existed in that time period. It is also easier to use shorter pieces when wrapping the body. For a separate piece to be used around the head would not be difficult to understand when considering the practical challenges of wrapping the body.
The difficult question is definitely why did Jesus fold the burial cloth that was around His head? There has been some speculation that this is related to a circumstance when a master of a home is dining. It has been suggested that it was a custom that the master would fold his napkin if he intended to leave the table for a short time, and then return to continue eating. I assume you are referring to this suggestion. I am not aware of any specific historical evidence that this is true. There might be something, but I have not yet found any evidence to this claim. There is no reason given in the Scriptures, for why Jesus folded the cloth. There may be no special significance, and people are making assumptions about what the significance may be. Without a specific reference or citation to the claim, it is necessary to consider it with some skepticism.
If someone was dining at their table, and someone else was going to clean up after them, an indicator that the person was finished would be helpful to ensure better communication between the master and the servant. The napkin would be a very good indicator that the master and servant could use to ensure the servant doesn’t clean up until after the master is finished.
There are some discussions concerning how you handle your napkin when dining, found in the Talmud. For example, in Berakoth 52b there is a debate recorded between the house of Shammai and the house of Hillel. According to Shammai, if you wiped your hands with your napkin, you were to place the napkin on the table, not on the seat cushion. The reasoning was that the napkin could become unclean due to having contact with the seat cushion, and then defile the table and the food on it if it came in contact with the table again. According to Hillel, you were to place the napkin on the cushion and not on the table. The reasoning was that the table could be unclean already, and therefore render the napkin unclean, and any food the napkin might come in contact with. The reconciliation between the two opinions was that each should be careful with what they decide to do.
The reconciliation between the two houses of Rabbinical thought was to be careful with how they handled their napkins. There is no hint of folding or not folding napkins in the discussions. Those who were of Hillel would not have folded their napkins and placed them on the table. They might have folded their napkins during or after dining, but they would have placed them on the cushion of their chair. Those who were of Shammai would have probably folded their napkins before placing them on the table, to try and keep them from coming in contact with something that is unclean. If a person was finished dining, then it wouldn’t matter how they handled their napkin anymore. If the person dining was following the teachings of Shammai, they would probably have to make a point of not folding their napkin to indicate they were finished eating. If a person was following the teachings of Hillel, a napkin on the table would be a very exaggerated way of indicating that they were finished eating, whether is was folded or not. The folded napkin would have no significance according to the teachings of Hillel. The significance would be if it was on the table or not on the table.
When attempting to make a correlation with Jesus’ folded cloth, the important thing to understand is that there was no table involved. The Rabbinical debates were over the condition of the table or the cushion, not whether the napkin was folded or not. For this reason, I think it would be a very far stretch to consider any significance to the cloth being folded or not folded. If Jesus was giving a message that He was coming back because the cloth was folded, it would have been hard to identify. Those following the teachings of Hillel did not require the napkin to be folded or unfolded, but would see that it was resting on something that was considered to be a seat. I believe that would be the extent of what the disciples would observe given the limited knowledge available.