I was able to find a study that looked at this. They looked at synonyms in a single thesaurus:
The synonyms of a headword w are whatever words are listed in the entry for w in an on-line version of The New Collins Thesaurus (1984) (CT). According to the authors, "...no synonym is entered unless it is
fully substitutable for the headword in a sensible English sentence"
The paper reveals that the thesaurus has a lot of asymmetry:
62% [of synonymy links] are asymmetric (e.g., part has department as a synonym, but department does not have part)
They found several reasons for this asymmetry, most of which can be chalked up to omissions. However, a substantial portion of the asymmetry (18%) was due to hypernymy:
For example, book lists manual as a synonym, but manual does not list book; instead special types of books such as handbook are given. This is because book is really a hypernym (not a synonym) of manual. Hypernym links are truly asymmetric in nature.
A Tool for Investigating Tile Synonymy Relation in a Sense Disambiguated Thesaurus