I think it's clearer to create a tree:
Empathy
|--Sympathy (positive view of the result of empathizing)
|--Antipathy (negative view of the result of empathizing)
For example, one could empathize with someone who is consumed with hatred of other people, and possibly understand how they came to adopt that view, while strongly disliking that kind of mentality, and that would be antipathy. One could even have compassion for such a person, while also accepting that they need to be locked up in a prison because of their history of serial murder of other people.
Alternatively, one could empathize with a parent fleeing a war zone with children in tow, and most people would be sympathetic towards that person and their motivations, because one could understand why a parent would want to protect their children from harm, at a personal emotional level. A compassionate response would be to say, "Perhaps our government / community can provide some basic resources for these people, and why not also try to stop that war from continuing so we don't have to take in all these refugees, as well?"
Empathy doesn't really have an opposite, but it is a capability which may be missing in some people, comparable to vision vs. blindness. A person lacking any sense of empathy is somewhat rare, but they do exist and are generally categorized as sociopathic narcissists... they do tend to be a bit on the power-mad side of things, and hence are more frequently found running governments, corporations, and (other?) organized crime cartels, as well as starting wars, etc.
Perhaps the lack of empathy is genetic in nature? Someone could come up with a DNA test, perhaps.