tilly,
Yes, this is exactly what I was guessing at. There are also constraints as to what x, y and z can be within the confines of this puzzle. I didn't have a chance last night to work on this though because the following entered my mind as I was leaving work
X+Y X+Z XYZ
X2YZ X X+2Y+Z
X+Y+Z XZ XY
Terms: X, X+Y, XY, X+Z, XZ, X+Y+Z, XYZ, X2YZ, X+2Y+Z
X+Y X2Y+Z X+YZ
XZ X X+Z
XY+Z X+2YZ XY
Terms: X, X+Y, XY, X+Z, XZ, X+YZ, XY+Z, X+2YZ, X2Y+Z
Terms in common: X, X+Y, XY, X+Z, XZ
Terms not in common: X+Y+Z VS X+YZ AND XYZ VS XY+Z
X2YZ VS X2Y+Z AND X+2Y+Z VS X+2YZ
I haven't convinced myself that you don't need to iterate over other series of equations. Still thinking on it though.
Update: In fact, I think I can show that your statement "By rotating and reflecting we can make the largest corner be x+y" is not in fact true  consider the following square:
X+Y XYZ X+Z
XY+Z X X+YZ
XZ X+Y+Z XY
You have X+Y and X+Z both in a corner so you need to make a relationship between them to determine which is the largest value. Also, I noticed that I showed it is possible to have X+Y+Z in a corner and am now convinced that multiple series of equations must be iterated (though you can reuse the values for X, Y and Z).
