I don't know what type of toilet mounting flange the new rigs have, but here was how I could move mine.
Unscrew the toilet mounting nuts. Lift up the toilet. Move the mounting bolts to the next slotted holes in flange, if available. If not, remove the screws that hold the flange to the floor. Rotate the flange (it screws onto the tank). If it is too tight to rotate, you may have to make a tool from a 2 x 4 and two lag screws to get enough leverage.
Your rig may differ from mine.