My mother told me a month of so ago that I can't possibly be an aspie because I'm the most compassionate person she knows. Whilst hearing it did give me the warm fuzzies inside, it did get me thinking. I believe that as with everything else aspie-related, it's a spectrum.
There are two components to empathy as experienced by other people. There is the internal feeling of empathy, and there is the communicating the empathetic feelings with others. By the very nature of aspergers, an aspie is going to suck at the second part, even if they are very good at the first.
At one end of the spectrum, you have aspies who cannot empathise with another person at all. They just can't understand the idea of another person having legitimate emotions. I suppose this would be the Theory of Mind waffle that's so popular.
At the other end, you have aspies who feel empathy with others strongly, but due to their aspie nature, they have extreme difficulty with communicating that empathy with other people. They might empathise strongly with whatever pain someone else is going through, but if they lack the social skills to 'read' what is needed in the situation, they're going to do or say something inappropriate, and others will come to the false conclusion they lack empathy. After all, empathy is the condition of feeling what someone else is feeling, and treating them the way you would like to be treated in the same situation.
An aspie who would like to be left alone to deal with things and hates to be touched is going to provide someone else that same emotional distance if they see the other person is upset. But the NT will interpret that distance as lack of empathy and be offended, because what makes the aspie feel better is not what makes the NT feel better.