Jigsaws are projects that we frame from time to time. The jigsaws must be dry mounted onto a thicker piece of card so that they remain flat (and in one piece!) throughout the framing process. A customer bought in their jigsaw to be framed in an ingenious roll up storage device that you can easily transport the jigsaw without any concerns about the jigsaw falling apart. The idea is that you work on the jigsaw on top of the cloth.
Then when you have finished working on it you can simply roll the jigsaw up until you want to work on it again.
This must also have the advantage of freeing up the dining table whilst the jigsaw is being solved! I suspect this is a great idea that has been around for a while and I have only just stumbled upon it but it certainly beats the splintered piece of marine ply that I used to use for jigsaws!