The problem with this logic is that you're not applying it correctly.

But if we can have, say, "2x more detergent" then can we not also have "2x less detergent"? 2X more detergent would be 200% the original amount of detergent, so wouldn't the inverse also be true?

Let's say you have 1l of detergent. What's 2x1l of detergent? 2l of detergent. If you say 2x more, you have to add it to the original amount*. So 2x more detergent than 1l would be 3l, thus 300%.

This is also where it becomes apparent that the inverse can't be true. 2x less than 1l would be -1l and I haven't seen that being sold anywhere.

What you described is 2x as much/many, which, however, doesn't work inversed either. You'd say half of, quarter of or x y-ths of something.

So the problem here really isn't what x times less than something is, but rather that it just doesn't make any sense if you're not working with negative numbers, since as long as x is >=1, you won't be getting anything above 0.

(* 12 more than 3 is 15, not 12. 12 is also 4x as much as 3. So if you say 4x more than 3, you basically say (4x3)+3, read: 4 times more than 3.)