Please see notes on chart.
The problem is, if you draw correlation, say, XAUUSD and DXY, it's hard to say that most of the time they are negatively correlated -- like people often say "...dollar up, gold down; dollar down, gold up...' -- the correlation changes the sign over time.
What you are doing through looking at 1/DXY and XAUUSD correlation -- you are basically inverting dollar index. Because Dollar index is mostly dominated by EURO, one can use EURUSD in place of 1/DXY, so that inversion mechanism is more or less obvious. But as I previously wrote, I'm unable to find persistent (anti) correlation of currencies and commodities.
Basically my point is that inverting DXY doesn't give any new information in terms of correlation. Without inversion you get one sign, with the inversion -- another.
More important point is -- is it possible to say that in certain situations correlation works for some reason, and doesn't work in others?