>Agglutinative languages, like Finnish or Hungarian, where word order is largely irrelevant and conjugations, not word order, determine meaning.
I'd argue that word order is just as relevant (and indeed constrained by meaning) in Hungarian or Finnish as it is in, say, English, though the details of how each language's syntax works are very different. Also, there are plenty of languages not usually characterized as "agglutinative"(e.g. Russian) in which affixes rather than word order are the primary determinant of the role that a word plays.
"Agglutinative" is really just a way of saying that meaning-bearing morphemes in the language tend to be more closely bound together: where in English, for example, we'd express "in our houses" with 3 words, whereas in Hungarian, these would form a single word "házainkban" (ház[a] "house", -i- plural, -nk "our," -ban "in"). We know they're separate words in English because other stuff can come between them ("in each of our big houses"); not so in Hungarian. (So arguably agglutinativity is more about constrained morpheme order than it is about free word order!)
>Topic/subject languages, like Japanese, where previously defined context can radically alter meaning. "Watashi wa hamburger desu" can mean "I'm a hamburger", "I'll have a hamburger", "If you ask me, it's a hamburger", etc.
Of course context can radically alter meaning in English too; think of the waiter who says "which one of you is the hamburger?" ("Me, I'm the hamburger, and he's the Caesar salad.")
Almost all language classifications exist along a spectrum, or perhaps it'd be better to talk about a high-dimensional space. Different languages rely more or less on different strategies, but there are almost no linguistic features for which analogues can't be found in most languages.