This is mainly a philosophical problem: what are opposites?
You can only have opposites if you have a frame of reference. In one context, the opposite of man can be woman; in another, it can be beast or nature or child, and so on. Metaphorically speaking, you need a mirror to reflect against, or a hinge to pivot around. You need to reduce the number of options to two for there to be an opposite.
In many cases, using a negation can be an obvious way to do this, because you basically divide the entire spectrum of possibilities into x and "not x": unfriendliness can be the opposite of friendliness, and unloved can be the opposite of loved. It could be said that a negation provides a mirror. So you could say the opposite of Schadenfreude is not Schadenfreude, which would mean simply the absence of Schadenfreude.
However, the more specific a word or concept is, the more other possibilities you are grouping under "opposite" by simply using a negation: there are many kinds of not-Schadenfreude, you could say. That is why it doesn't seem a very satisfying kind of opposite in this case.
We could instead decide to focus on the aspect joy, and mirror it along the spectrum of "good" versus "bad" emotions. Even then, there seem to be two options: anger and sorrow. You could be sad about someone else's misfortune, or angry. So the spectrum of good v. bad emotions is not even specific enough. Both could be used as opposites.
The aspect of misfortune could be used as well, and we could choose the spectrum of good versus bad things happening to someone else. Then the opposite good fortune would be fitting, so you could be happy about someone else's good fortune.
What we did here was pick just one element of meaning expressed by the word and find some frame of reference in which to mirror it. If you take several elements at once and mirror them each, like feeling angry (or sad) about the good fortune of others, then you have less of the original meaning left (neither joy nor the other's misfortune), and so it is less recognizable as an opposite. But you could still consider it an opposite.
No doubt other opposites could be created, such as eh...the most commonly used English word used in German (could be cool), assuming that Schadenfreude is the most commonly used German word in English. You could say that is an opposite as well; it is just based on very different elements, namely the frequency of the word, its original language, and the language we are using it in now. Possibilities are endless, and none seem categorically different from others. For that reason, there are no meaningful words for different opposites: each word just has different options.