I will ask my question through an example game:
Each round we are given a blue coin and a red coin. We are either given a pair of fair coins or a pair of biased coins (these four coins are the set of all coins; there are no other coins).
For some valid reason, we have high confidence that these two biased coins are biased in almost the same way.
Let there be some small number $n$.
We go through $n$ rounds and observe the outcomes. In a particular round for example, the blue coin and red coin could land heads and tails respectively.
When we are trying to estimate $f_{X, Y}(x, y)$, does it make sense to add "mirrored" points to the sample data? In other words, for each pair of blue and red flips, should we add a "mirrored" data point where we say the result of the blue flip was what the red flip actually was and we say that the result of the red flip was what the blue flip actually was?