We have received a very clearly explained solution to this, but unfortunately whoever sent it did not include their name. If it was you, let us know!

Let ab=x and cd=y (where ab means a as the tens digit, and b as the ones digit, not a times b).

abcd2-cdab2 ab2-cd2 = (100x+y)2 - (100y+x)2 x2-y2

= (10000x2+200xy+y2) - (10000y2+200xy+x2) x2-y2

= 9999x2 - 9999y2 x2-y2

= 9999(x2-y2) x2-y2

= 9999

(since x>y we are not dividing by zero)