The geometry for a particular puzzle is defined in a file with the extension ".sym". This is one of the Foldit_file_and_directory_structure#Puzzle_files puzzle files that are downloaded from the Foldit server and reside in the Foldit installation directory.
Unlike the other puzzle files, the .sym file does not have a digital signature, and so it appears to be modifiable.
As an example, the puzzle Symmetric Design with Filters has a file 0000996730.ir_puzzle.sym, which looks like this:
symmetry_name c3z E = 3*VRT0 + 3*(VRT0:VRT1) anchor_residue 41 virtual_coordinates_start xyz VRT0 0,0,1 +0.0000000,+1.00000,0 0,0,0 xyz VRT1 0,0,1 -0.8660254,-0.50000,0 0,0,0 xyz VRT2 0,0,1 +0.8660254,-0.50000,0 0,0,0 virtual_coordinates_stop connect_virtual J0 VRT0 VRT1 connect_virtual J1 VRT0 VRT2 connect_virtual JS0 VRT0 SUBUNIT connect_virtual JS1 VRT1 SUBUNIT connect_virtual JS2 VRT2 SUBUNIT set_jump_group JUMPGROUP JS0 JS1 JS2
The easiest part is the symmetry name, c3z, which probably translates to C3(z). The "C" is for "cyclic", referring to something that forms a ring. The "3" refers to the number of parts, means this is a trimer (see below). A cyclic has only one axis of rotation, in this case the "z" axis.
The bottom line is that this puzzle has three units. Moving the main chain in one direction moves the copies in the same direction. Rotating the main chain rotates the copies in the same direction (each from its own perspective).
The second line
E = 3*VRT0 + 3*(VRT0:VRT1)
seems to be defining the identity operation for the puzzle. (The "E" is from the German Einheit, meaning oneness or unity.)
The exact meaning of the identity equations and the rest of the .sym file will be worked out in future updates.