Monday Mini-Golf #3: Anagram distance
Pyth, 12 bytes

ls.-M.prR0.z


Test suite

The operation in question is equivalent to Pyth's bagwise subtraction operator, .-, applied in both directions. You could call it bagwise xor, I suppose.

The solution is:

.z: get input as list of 2 strings.

rR0: convert both to lowercase.

.p: Form all permutations, i.e. normal and reversed.

.-M: Map the .- operation over each ordering.

s: Concatenate the results.

l: Print the length.