Hi Valerie,
Yes, the Rosetta group has two basic patterns: simple rotations (restricted to angles that evenly divide into 360) and rotation plus reflection - I didn't see the reflection option in your design, perhaps I missed it.
I have been thinking about how the properties of a symmetry group can be visually represented so that the user can understand what is going on (without studying Group Theory or cracking open a textbook on Symmetry). Below are three examples where I have tried using annotations in the diagram to represent how the wallpaper group symmetry will be applied to the base tile:
http://inkscape.13.n6.nabble.com/file/n4631649/3xsym.png
The types of actions performed on the base tile depend on the wallpaper group (of course). Sometimes at the intersection of the base tile lines you have a rotation. Sometimes along a line you have reflection or you may have a glide reflection (reflection plus translation). So I was thinking that in addition to having lines which define the size, position and orientation of the base tile, the guides could provide information about what will happen along a line or at a corner. In the diagram above I have used little propellers to show that a rotation will occur at a corner as well as a dashed lines to show the edges that define the rotation angle. I have used a little barbell symbol and a dash-dot-dash line to indicate an edge along which a reflection will occur. I am still thinking how best to represent a glide - I will have to get back to you on that one.
In some cases, the user will want to flip the guides (vertically, horizontally) in order to align with the directions of the wallpaper actions with the directions of their design. I don't have an example of this in my diagram but it could occur for example with 4*2 (also known as p4g).
Are your guide points a type of marker to show the user where a given point will appear (with orientation) under the tiling operation? Can the user change the markers in any way or are they for information only?
Seeing a shadow of the operations (see the second column of the diagram above) could also be very useful at design time - it gives nice feedback about how a design behaves at the edges without having to generate the entire tiling. If the shadow is limited to one example of each operation it may not place too high a demand on performance.
I am no expert on Symmetry Group Theory but I am very interested in learning more about it and understanding and teaching Symmetry concepts to non-Math people and students just getting started in Math.
Cheers, Veronika
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