And to follow on from BM's answer.
While you are talking to them explain that a sky dish is a receiving antenna only, it does not transmit anything, so how close they are mounted to each other doesn't matter at all as long as one is not in front of the other therefore blocking the rear dishes view of the satellite they could be mounted touching each other.
You say yours is the upper of the two dishes so it is possible (though very unlikely) that there is an obstruction in front of the pair of dishes that is blocking a clear view for his dish but yours can just see over the top of it.
it is far more likely (as I said above) that the connection onto his dish was not properly sealed against water when it was installed (very common problem) and water has seeped into the end of the cable. This is an easy problem to resolve, just cut a few centimetres off the cable and refit the connector, wrap well with tape to waterproof and attach to the dish again.
I have to say I like BM's reply above, and if he is correct then you would indeed by entitled to ask them to remove their dish as it overhangs your property, also if they want to get any one in to look at their dish it sounds llike he will have to ask your permission to access it via your property.
I think your neighbour is going to be causing more problems for himself if he continues being petty about where your dish is mounted, I just hope he can see sense and realise that your dish could not cause interference to his and he gets whatever problem he has with his dish sorted rather than blame other people.