That you can track VOR in that way with an HSI is something of a double fluke!
If you fly towards a VOR with a conventional CDI with the OBS correctly set (magnetic track to the VOR), indications will be correct. If you fly towards a VOR with a conventional CDI with the OBS set 180 degrees off (magnetic track from the VOR), indications will be reversed.
Try the same with an HSI. In the latter case, the indications are still reversed. But the left and right indications are with respect to the direction of the course arrow. Because the course arrow is now pointing downwards, the entire readout system will be upside down. Thus the readout is reversed twice, and reads in the correct sense.
For a localizer, the direction of the course arrow is irrelevant. You can fly the ILS with anything set on the OBS of a conventional CDI. However, if you set the course arrow of an HSI to 180 degrees off the localizer, the course arrow is now pointing downwards and the readout system is reversed -- just once now -- so the instrument reads in reverse.
The autopilot, of course, doesn't care which direction the course arrow is pointing. So if you try to fly towards a VOR with the course arrow set 180 degrees off, the autopilot still sees reversed indications and fails to track the VOR.