Counting inversions is equivalent to sorting. That's why it's 0(n log n). Any sorting algorithm can become an inversion counting algorithm by incrementing a counter instead of rearranging the input values. Inversions are the first topic in The Art of Computer Programming Chapter 5. Very much worth reading if you dig this stuff.
Integer sorting seems like a case where there's meaningful metadata (maximum integer) and the algorithm is written only for that case (and assumes suitable hardware). These aren't radical assumptions and the optimization may have a lot of practical applications. But once I get to specify the inputs, then O(1) sorting is trivial. Specify the input as sorted lists. To me, it only seems a matter of degree. But admittedly, what can be represented as an integer is a much more reasonable degree of specification.