I'm trying to recall the term for an operator/function where
f(a, b) = f(b, a). For example,
a + b = b + a for all values of a and b. However
a - b != b - a unless
a = b.
I think the word you are looking for is commutative:
(mathematics, of a binary operation) Such that the order in which the operands are taken does not affect their image under the operation.
Addition on the real numbers is commutative because for any real numbers s , t, it is true that s + t = t + s.
Addition and multiplication are commutative operations but subtraction and division are not.
You seem to be asking for the word for operations and functions. The word mathematicians generally use is different in the two cases.
A function which does not depend on the order of the arguments is usually called a symmetric function. So f(a,b,c) = f(b,a,c) = f(b,c,a), although the word commutative is occasionally used.
And as the other answers say, an operation which does not depend on the order of the operands is called a commutative operation. So a·b=b·a.
Any operation ⊕ for which a⊕b = b⊕a for all values of a and b. Addition and multiplication are both commutative. Subtraction, division, and composition of functions are not. For example, 5 + 6 = 6 + 5 but 5 – 6 ≠ 6 – 5.
As others have noted, the common term for this is "commutative". For a brief etymology explaining why the word for this is "commutative", see https://math.stackexchange.com/questions/3045924/why-is-it-called-commutative-property
You might also hear "Abelian", named after the mathematician Abel, who studied mathematical structures called "groups" with commutative operators in the 19th century.
FYI there's a hilarious joke about that which will help you remember this fact:
What's purple and commutes? An Abelian grape.
See https://en.wikipedia.org/wiki/Abelian_group for more details.