BIK, your last sentences are wrong. There is NO application for setting back course LLZ track in the COURSE setting of an HSI. Doing so will reverse all the indications of the LLZ. The HSI will always indicate correct displacement towards or away from the LLZ irrespective of heading or position, provided the front course is set.
To any wishing to understand how to fly an LLZ on HSI whether inbound or outbound, on the front course or the back course, there is only one setting EVER for the HSI COURSE setting - the front course track. That is, the runway QDM associated with that ILS. (And I am cognisant of the twaddle going on re that title on another thread.)
Vandelay, nobody seems to have answered your question as to why the raw ILS indicates in reverse when inbound on a back course, and why it can't be cured by the OBS. Going right back to basics, if you care to consider a runway due north/south, with a LLZ on the 360 inbound, there is a signal which predominates to the west of the field and which deflects the LLZ indicator to the right side of the dial, and another signal predominating in the east which deflects the indicator to the left. If you are heading north, turning towards the needle will take you towards the centreline. (This will apply whether you are north or south of the field, if there is a useable back course - it is the same signal.) If you are heading south, and still off centreline and to the west of the field, you are still in the same signal area and the needle remains on the right side of it's dial. However, the runway, on this heading, is now on your left. Hence having to turn away from the needle to close.
As said, the OBS has nothing to do with ILS, and the only cure for this reverse command is a BACK COURSE switch found in some aircraft for ILS indicator, Flight Director or autopilot. This reverses the display and inputs to enable the instinctive tracking desired.
When it comes to the HSI, the COURSE setting (which also operates the OBS input when the HSI is used on VOR) has no electronic effect on the display of ILS, but it orients the LLZ indicator which is mounted in the centre of a compass rose. Now the front course setting has correctly oriented the needle and if you fly a back course the compass will have rotated the display so it will now indicate as desired. If that gives you trouble, try to imagine the HSI mounted vertically with it's face horizontal - you now have a geographic display of your whereabouts in relation to the runway. Sufficient imagination will show how the display reacts in a 180 degree turn, where the above case of changing direction remaining on the same side of the runway now rotates the needle to the correct side of the aircraft. There is further advantage in that the orientation as well as displacement of the LLZ is apparent at all times, irrespective of heading.
Indeed the Collins (I think) GA form of HSI was the PND-101 which title legend suggests stood for Pictorial Navigation Display.
VOR radials also make much more sense on a HSI but that's another story.