Orientation-reversing involutions of the genus 3 Arnoux-Yoccoz surface   and related surfaces
  We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square. 
Joshua P. Bowman