but when 155 and 105 are XORed, that produces 240, not 242.

Says who? The exclusive or of two bit patterns contains a one for each mismatch pair of bits and a zero for each matched pair. Thus for 155 and 105:

# 1001 1011 - 155
# 0110 1001 - 105
# 1111 0010 - 242
[download]

The fun thing about XOR is that if you XOR the result with one of the input values, you get the other input value back:

# 1001 1011 - 155
# 1111 0010 - 242
# 0110 1001 - 105
[download]

Thus various encryption schemes (and some fun graphics programming on early PCs :)