Polycairos are shapes formed on the Cairo grid composed of identical pentagons with axial symmetry. The cells are grouped four by four in irregular hexagons. The pentagon itself can have slightly different proporitions but always has two angles of 90° and four sides of the same length.
There are 5 tricairos, 8 one-sided tricairos, 17 tetracairos and 31 one-sided tetracairos.
What can be done with the tricairos? Not much, indeed. Here is the only symmetric shape that I could find by randomly clicking around in my solver:
Since I was not successful in finding other symmetric shapes, I decided to try some symmetric holes:
Since 8 × 3 = 6 × 4 one could hope to tile six hexagons with the one-sided tricairos. This turns out to be impossible for parity reasons. Same problem with the 31 one-sided tetracairos which cannot tile 31 hexagons. However, if we combine both sets, it becomes possible to tile 37 hexagons. But 37 is a centered hexagonal number!
Many other symmetric shapes can be produced following the same idea.