If "x" and "y" are terms for things, "(x,y)" and "(y,x)" are terms for pairs. Note that the left element of "(x,y)" is "(x)" while the left element of "(y,x)" is "(y)". Hence (x,y)=(y,x) only if (x)=(y). Here in either case the parentheses are merely interpunction, like the commas.

The point is not the terminology but that one can distinguish the two elements somehow by their place in an order.

In standard set theory one can analyse pairs as sets of a certain kind: The pair (x,y) becomes identified with the set {{x}, {x,y}}, which has the same property as distinguishes a pair: One can keep apart the left and the right member, except if they are identical.

Norbert Wiener first thought of this. It is an elegant reduction and standard practice in set theory, but it has the disadvantages that (1) it requires sets of sets to work and (2) it will not hold for proper classes, which are not sets. For this reason e.g. Cech, in his "Topological Spaces" chooses to work with special axioms for pairs, that also hold for proper classes.