Two polyominoes are compatible when there is a figure that can be tiled by both; a smallest one is called a least common multiple.
In looking through tables of these least common multiples of various polyominoes, I noticed that a vast majority are symmetric. (See http://www.recmath.org/PolyCur/pairwise.html and http://www.recmath.org/PolyCur/pairwise.html).
Could someone explain why this is?
Not all solutions are symmetric, here is an example:
(Image from http://www.iread.it/Poly/P65.html)
(Image from http://www.recmath.org/PolyCur/n8euph/n8euph.html)