I tried to draw PT (Penrose tiling) starting by a small wide rhomb and going upward in hierarchy. See the following picture where I started from the small light green rhomb and proceeding one hierarchy up at a time using different hues of greens.
I found that it is possible to tile the entire plane using this method and it has the following properties:
1) There is a reflection symmetry along the straight line passing along the longest diagonal of the starting rhomb. In this picture the symmetry is obvious between the left upper and the right lower sides.
2) There is a straight line (red line along the diagonal in the picture) going to infinity and passes along the longest diagonal of the starting rhomb.
3) It is possible to re-tile the plane going 1-hierarchy up and still the reflection symmetry is preserved.
4) Re-ltiling 1-hierarchy up like in (3) results in a patch which is a scalled version of the starting patch but with reflection along the perpendicular direction of the red line that passes through the longest diagonal of the starting rhomb.
I am concerned then that Pnerose Tiling has this weired reflection symmetry not just 5 rotational symmetry which is known for.
Also if this is true, then there must be a straight line that goes to infinity across the entire plane and this line passes along the longest diagonal of one of the wide rhomb. In other word, if you give me an arbitrary penrose tile, there may be a way to find a rhomb and a line that passes through the longest diameter of that thomb and goes to infinity and divides the plane into two symmetrical halves.