Many modern linguistic analyses of languages like English draw a sharp theoretical distinction between participles, which are analyzed as inflected forms belonging to the paradigm of some verb, and deverbal adjectives (also called verbal adjectives, participial adjectives, or departicipial adjectives), which are seen as adjectives derived from verbs. (There are certainly disputes about the validity of the inflection/derivation dichotomy, but it nevertheless seems to have wide acceptance. There are also disputes about whether participles should be analyzed as verbs, adjectives, neither or both, but I don't think that's as relevant to my question. Also, the word "participle" is often used in a broad sense that covers both an inflected form and a derived word with an identical form, but the broad vs. narrow definitions of "participle" just seem to be a matter of terminology.)
However, it can be tricky to determine which category a word belongs to in a specific example sentence. There are some syntactic tests, but often they only work in particular situations and can only give a definite answer in one direction. (E.g., in English, if an -ing word has a direct object, that is considered to be proof that it is a participle and not a deverbal adjective, but if it doesn't have a direct object, it might be either a participle or a deverbal adjective. If an -ing word is modified by the word very, that is considered to be proof that it is a deverbal adjective and not a participle, but if it is not or cannot be modified by the word very, it might be either a participle or a deverbal adjective.)
I would like to know about any such syntactic tests that have been noted or proposed for distinguishing Latin participles (in the sense of "inflected forms") from derived adjectives.