Meaning if one word (word a) is a synonym of another word (word b), does that automatically mean word b is a synonym of word a?
For example, in the oxford dictionary "straw" is listed as a synonym of "hay" but not vice versa.
Do they just not list all words? Or are synonyms not commutative?
It's practically an article of faith in linguistics that there are no perfect synonyms. There are lexical items that have virtually identical meanings, but they're never quite identical enough to substitute for one another in every context. It's easy to find contexts where one lexical item generates an ungrammatical sentence, but a "synonymous" word doesn't.
As for commutativity, that's a mathematical concept that has to do with order of operations: 2 + 5 equals 5 + 2 because addition is a commutative operation, but 2 - 5 does not equal 5 - 2 because subtraction is not commutative. There are commutative phenomena in English, like the following three constructions with marry, which are all grammatical, and all mean the same thing.
But synonymy itself is not not subject to commutation; it's not a construction.
One could consider marry and get married to as synonyms; but one's an idiomatic construction with an inchoative and a preposition, and the other's a transitive verb. They can be used in different contexts, which is what's really useful.
synonym = a word or phrase that has the same or nearly the same meaning as another word or phrase in the same language
If two words are perfect synonyms their meanings comprise two perfectly overlapping sets of concepts. In this case they commute in the sense that you have chosen to use it (not the strict mathematical sense. See footnote). For example, we might consider fearless and brave.
But note: "or nearly the same" in the Cambridge definition.
The overlapping parts of the meanings commute, but the small parts of each meaning that are unique to each word mean that commutation of the whole words is impossible.
In most cases (as in the Cambridge definition), the synonymity is not quite perfect, in which case each word may have some concepts additional to or not shared with the other.
In consequence it may be possible to compose a progressive list of synonyms in which small areas of non-agreement accumulate to the point that the final word bears little relation to the first. Her is one such list pulled from synonym search: Clever; Smart; Sharp; Intense; Harsh; Brassy; Bold; Unafraid ...
Footnote: in the mathematical sense, if the order in which two things are done has no effect on the outcome, the operation is said to be commutative (multiplication: 2x3 = 3x2). If changing the order changes the outcome, the operation is non-commutative (division: 4/2 ≠ 2/4).
As a rough parallel, synonyms commute in the sense that you use the word, in that one may replace the other in writing with no effect on overall meaning.
If things do not commute (I drowned then died is not the same as I died then drowned), their exchange alters the meaning. Thus it may be with imperfect synonyms: if exchanged the meaning of the prose is altered.
