Is a finite tiling of 45-45-90 triangles uniquely determined from the resulting union (up to trivial flips)?

Question

Suppose I take a union of nonoverlapping $1-1-\sqrt{2}$ triangles in the plane:

The same shape can be tiled in another way, by flipping two triangles joined together into a square:

In general, are these the only such decompositions possible? That is, given two planar configurations of finitely many nonoverlapping $1-1-\sqrt{2}$ triangles whose unions are the same, are they necessarily related by some number of "square flipping" operations?

In the infinite case, this is not true, as seen by the following tilings of an infinite quadrant:

Answer 1

While it feels a little like cheating, the following configurations seem to provide a counter-example to the stated conjecture: