The Veech group of this surface contains at least two transverse pseudo-Anosov elements (shown by Arnoux-Yoccoz and Hubert-Lanneau-Moller). The concentric circles in this picture are images of each other by the original pseudo-Anosov element, which fixes the horizontal and vertical directions.
A detailed description of this surface can be found here.
The next pictures are rotations of the above, around each of the two kinds of vertices along the vertical axis.
The rotation sending one of the concentric circles to the vertical axis:
The other rotation:
A closer view, for more detail:
It seems that the horizontal and vertical directions in this rotation should be periodic, but it's not clear from the above. The following picture proves it, by zooming out sufficiently:
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