wvmarle:
Works for any A where B = 10 (and every B where A = 10).
I could not sleep the other night and thought about this - it did not help with the sleep 
AB=BA it's algebra Jim, but not as we know it. The "trick" works because although the 'B' on both sides of the equation look the same they actually have different numeric values as they are in different bases.
Any base written in that base is always 10. As @wvmarle wrote the trick always works as long as at least one of the subscripts is 10. However it works whatever base the subscript is in as long as it is 10 e.g.
Binary: 1110 = 1011 = 3 decimal
Octal: 1110 = 1011 = 9 decimal
Hexadecimal: 1110 = 1011 = 17 decimal
Hexadecimal: 1F10 = 101F = 31 decimal
Curious, but I am not sure what use it has.