Suggestions:
Full notation: write the position of all corners.
Short notation: write the position of the sharp corners clockwise; use any two adjacent corners for the square, start with the parallelogram if the long edge follows the sharp corner, otherwise end with it. (Note that the short notation requires that the reader knows the 2 rules, and the square and parallelogram can be written in 4 resp. 2 valid ways).
Example (top left): {2,3;4,4} {2,2;0,0} {0,4;2,2} {1,3;3,3} {1,3;0,4} {3,4;2,3} {1,3;1,4}
(Note that in the short notation, the interpunction is not needed. But it makes it much more readable)