I've never considered love and hate to be opposites, since they can co-exist in one's regard for something. Doesn't one both love and hate their parents from time to time? They're not mutually exclusive so, to me, they're not even the same coin. Love derives from affection, hate derives from anger ... and I'll stop that progression right now before I start sounding like Yoda.
To me, the true opposite of both love and hate is ... apathy