One thing I love about xor is an interesting correspondence between bitwise xor and the outer product of the exterior algebra.

Say we have an N=4 dimensional vector space, and we use binary place values to represent units vector in a basis, like this:

w = 1000

x = 0100

y = 0010

z = 0001

Then a multivector basis could be represented e.g.:

xyz = 0111

xy = 0110

wz = 1001

Now, if ^ is the outer product:

xy^yz = xz ↔ 0110 xor 0011 = 0101.

wx^yz = wxyz ↔ 1100 xor 0011 = 1111.

and so on.

If anyone has more information about this, I'd be really interested in seeing it!