DFC - I've just sat in my simulator and proved what I said was correct. The only way we can both be right is if there are different kinds of HSI, but I didn't think that was the case.
[Edited to add:] When on the LOC allow a Beam Bar displacement to develop. Wind the arrow round to the reciprocal of the ILS. The BB will travel with it and NOT swap sides as it would if you did the same thing on a Radial.