I am now rather cross that despite being in possession of the relevant information - in this case about how the cutting dies for the cardboard jigsaws that I solve are made - I completely failed to observe that there was a symmetry issue here. A sort of senior moment.
Cutting dies are made out of (say) 1cm strips of bendy steel with a sharp upper edge. In essence, the die maker lays parallel strips on the base board, sharp edge up and running from top to bottom, maybe an inch apart. The strips will include all kinds of wiggles reflecting the prongs and holes of the intended jigsaw, but we are talking of continuous strips. The strips are then fixed down.
The die maker then fills in in the other direction, but this time instead of a small number of long strips we have a large number of short strips of steel, one for each piece. And while the die maker may choose to line all these left-to-right strips up, as if they were a single, continuous strip, he does not have to. The short strips may be always aligned (in which case the jigsaw will have property 4), mostly aligned, sometimes aligned or more or less never aligned.
With the result that pieces are always aligned vertically but may not be aligned horizontally. Or vice-versa in the case that the die maker does it all the other way around. In the illustration, A is always aligned vertically with B, but may or may not be aligned horizontally with C.
So the puzzle has orientation. Top to bottom is different to left to right, a difference which can be exploited in solution.
PS: it would be interesting to see such a die. How do they fix the strips to the base board firmly enough to stand perhaps hundreds of cuttings? And how do you get steel which is hard enough to hold the necessary edge while being soft enough to be bent into the necessary shapes? The die has to hold the edge as I do not see how one could sharpen a completed die. Would it be worth taking one to pieces to sharpen it - then put it back together again?
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