Stewart Robert Hinsley
I'm not sure I understand the question:
is this question :"to jig saw or not to jig saw?".
This approach may be totally off from your question...
In any case is clearly delineates the difference between
fractal affine edged tiles and straight sided lattice tiles.
Suppose you have a square lattice tiling
and you replace one side with an "curve" or "arc",
then the jig saw effect is that all the symmetrically opposite
sides have to be made to fit as well with a second matching curve.
The projective plane-Wang tile set in a square lattice
makes the minimal square lattice "different" tile set
out at 13. Here they use 5 colors instead of curves:
https://en.wikipedia.org/wiki/Wang_tile
as five color on sides (or ten jig saw curves).
The curve option instead of the color one destroys the transformative symmetry options of rotation ( motion) or reflection.
What your question seems to be asking is if the transformation
symmetry group of a tile in invariant property of a tiling.
I have recently been doing work on polyhedral symmetry groups
which behave much like geodesic tilings of spheres
and the answer in this kind of case is that many polyhedrons can share the same group number of transforms while have very different polyhedral geometry. If you look at point groups in 3d, many share the same number of transforms, but use very different
transforms to form the geometry. The 17 basic plane straight line edge tilings and their transforms appear to
actually be "exceptions" or special cases when modern
tiling and space filling is taken into account.
But some polynomial based fractal tilings appear to be as you said "locked" and have very limited transformative ( rotational or reflection) options.
I know this isn't an answer, but maybe you need to be more clear about your meanings in your definitions.
--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)