By bidirectional I mean, if word A is synonymous with word B, does it follow that word B is always synonymous with word A? Are there any common exceptions to this rule?
Extending this - if word A is synonymous with word B, and word B is synonymous with word C, word C may or may not be synonymous with word A?
Perfect synonyms are equivalent, which means that all you say in your answer is true for them. But perfect synonyms are really unusual; some people say they don't exist at all.
In reality, synonyms are usually just partial, which means that they share a common meaning in some circumstances, but not in all of them. So the symmetric property (if A is a synonym of B, B is a synonym of A) holds true, but not the transitive (if A is a synonym of B and B is a synonym of C, A is a synonym of C).
Example: lead is a synonym of main (lead actor = main actor), graphite (in a pencil) or conductor (in electricity), but these words are not synonyms between them at all.
Part of the difficulty is what we mean by synonym. When we use reference works to search for synonyms, we are usually presented with a variety of choices. The Collins Thesaurus (found here) lists the following synonyms for the word pistol
- handgun
- shooter
- piece (U.S. & Canad. informal)
- automatic
- revolver
- side arm
Not all pistols are revolvers. Not all pistols are automatics. Revolvers are not automatics.
Very often a term that is either a subset or a superset of another term is considered a synonym in many contexts. One could argue that such offerings are not true synonyms, but in practice we may not make that distinction.
The very fact that we seek synonyms to words indicates that there are subtle differences between the word we know and the word we seek. If not, we would always use the original (except perhaps to avoid boredom or for euphony).
For beauty, the following are listed
attractiveness, appeal, charm, grace, bloom, glamour, fairness, elegance, symmetry (formal or literary), allure, loveliness, prettiness, seductiveness, gorgeousness, pleasantness, handsomeness, pulchritude, winsomeness, comeliness, exquisiteness, seemliness, pleasingness, prepossessingness
Some of these terms do seem to be true synonyms, but create different impressions or may be used in different contexts. Words that are listed as synonyms often have connotations that better suit a given context. While their denotation may be roughly equivalent or even identical, their shades of meaning are not.
Synonyms are words that sometimes have the same meaning, so they overlap. Even if A sometimes = B and B sometimes = C, that doesn't mean that A sometimes = C. See the 3-circle Venn diagram here.
You can think of two words like of tho sets of possible meaning (including denotation, connotation, contextual peculiarities, etc., etc.)
Thus, speaking of word comparison, we actually are speaking about set relations. Above-mentioned Venn diagram are nice visualisation of how actually sets can overlap.
So, no, not all synonyms are "bidirectional" - in mathematics this called bijective, or one-to-one correspondence.
pole -> post -> letterproblem is about transitivity, not symmetry. A relation can be symmetric, without being transitive. That is to say, it can be a rule that if A is a synonym of B, then B is a synonym of A. But that doesn't mean it has to be a rule that if A is a synonym of B, and B is a synonym of C, then A is a synonym of C. (But since synonymity is a kind of equivalence, it probably makes sense for it to be symmetric and transitive. Even reflexive: let A by a synonym of A.)