It's a fact that the grammar core of most European languages (not only IE ones) can be analysed in a relatively precise common framework. Of course I do not know much of these languages, but the basic concepts of nouns, verbs, adjectives, verb tenses, pronouns etc. seem to be relevant not only for Indo-European languages, but also for some of their neighbours (Finno-Ugric, Turkish, Semitic, Dravidian...) A perusal of a Finnish Grammar book, for example, will give to the IE reader an occasional surprise (vowel harmony, say), but won't transport him to another planet.
I'm not interested in a list of linguistic curiosities, but in cases where a serious grammatical analysis has been made that required a brand new look on the language. For example, a language grammar lacking a verb vs. noun contrast would certainly qualify.
Of course, while Lojban grammar perfectly illustrates the kind of exotic I'm after, I'm only interested in natural examples.
This is something like the whimsical companion of a more precise question I asked earlier.