I wouldn't consider hate and empathy mutually exclusive personally (e.g. you can empathise with somebody that was bullied by their parents, but still hate them

I wouldn't consider hate and empathy mutually exclusive personally (e.g. you can empathise with somebody that was bullied by their parents, but still hate them because they bully their children), but I do think that somebody that derives pleasure from the suffering of someone that they hate is unempathetic.