One can go to the other extreme and find suitable dissections of the square and the equilateral triangle without rhombs. For the square we get two different compositions of the sides. Thus we need two different kinds of triangles to join with the squares.
It seems that the resulting quasi-periodic tiling has not twelve- fold rotational symmetry:
Exchanging the dissections of the black and green triangles and adjusting the orientation of the squares we get a slightly different tiling. I am using a different scale than above:
We discover dodecagon units (regular polygons with twelve sides) made of six green triangles at the center forming a hexagon. This hexagon is surrounded by six squares and six black triangles. These dodecagon units arise in each iteration step at the corners of the triangles and squares.
Overall these tiling have only six-fold rotational symmetry because the dissection of the dodecagons destroys the twelve-fold rotational symmetry.