Beat Unknown Soldier at his own game of math.

[Unknown Soldier]

Here's the puzzle, restated briefly. I'll let Gospel fill in with any gleanings.

Puzzle: This 8-sized set ​ {Argentina, European Union, South Africa, Turkey} ​ UNION​ {Iran, Netherlands, Spain, Switzerland }​ is the Symmetric Difference between two other sets.​ What are those two other sets?

I expect Mr. Soldier to ask whether he should infer anything from the breakout into two 4-sized sets. I reply now that I am indifferent.

All the symmetric difference is is the set of elements in two designated sets that are not elements of both sets. You haven't posted enough information here for us to infer the two sets you are referring to. To see why, here's the definition of the symmetric between set A and set B:

A ∆ B = (A \ B) ⋃ (B \ A) = (A ⋃ B) \ (A ∩ B)

I prefer the (A ⋃ B) \ (A ∩ B) form. In English it says that the symmetric difference between sets A and B is their union less their intersection. So using your sets above one possibility is that the two "other" sets are

A = {Argentina, European Union, South Africa, Turkey}
B = {Iran, Netherlands, Spain, Switzerland}

Since A and B are disjoint sets, then A ∩ B is the empty set, and therefore A ∆ B is A ⋃ B.

Here's another possibility:

A' = {Argentina, European Union, South Africa, Turkey, Mexico}
B' = {Iran, Netherlands, Spain, Switzerland, Mexico}

Sets A' and B' are not disjoint their intersection A ∩ B = {Mexico}. Nevertheless, their symmetric difference A' ∆ B' = A ∆ B. So my point is that different pairs of sets can have the same symmetric difference. Knowing the symmetric difference of two sets A and B doesn't tell you what sets A and B are.