Perceptiveness is not empathy, or else we wouldn't need another word for it. Hatred and empathy are not on the same ends of the emotional spectrum. The ability to perceive others emotions and simultaneously carry toward them feelings of anger, hatred, or malicious intent is not empathy. It's called something else.
It sounds to me like your idea of empathy is more akin to compassion.
Yes, they go together. Sociopaths are low in empathy but excellent at perceiving the emotional state of others.