XOR HEX numbers without converting to binary?

[Killerbunny]

I have been thinking about making a hexadecimal computer. And I decided I want all my calculations be done using just the two comperetor functions combined with some fast memory. It all seems to be possible, until I think about XOR.
Hexadecimal XOR lookup table
I want help spotting patterns that can either solve the operation, or that it is simplified in some way. I really don't want to convert to and from binary so some help would be appreciated.

[Legofreak]

I just stared at it for a while and I think I might have an idea.
Each quadrant is a mirrored inverse.

Looking at one quadrant I see the numbers are mirrored across the diagonal. In that top wedge at 0,0 the output numbers are either a sum or a difference. I can't quite see the pattern yet.

When the number are equal they cancel.

When paired with 0 the output has no change.

[Killerbunny]

Yeah the mirroring was clear to me, but take a look at the 7 lines they follow a pattern I have seen before