With all of these questions, I am more concerned with the "why" vs. the right answer.
Nothing wrong with that!
The feathered arrow with shading is the indication it's a localizer. A localizer is an incredibly dumb device. It sends out a radio signal that allows the receiver to determine only whether it is on the localizer, left of the localizer or right of the localizer (and how many degrees).
In that diagram,
Everything on the grey shaded side will show left. Everything on the white shaded side will show right.
It's too dumb to show anything else.
You can spin the OBS knob on a traditional Instrument and the needle will not move on an ILS localizer. (Neither will the glideslope needle.)
You're essentially "undoing" all you've learned about an OBS from VOR work when you're learning to fly an ILS. Ironically this Instrument is one of the biggest breakers of the rule of primacy in the cockpit. You learned that rotating the OBS selects a radial of the VOR. Your brain wants to apply that information to the ILS and it doesn't apply anymore.
It's way easier to show this on a real HSI in flight. Especially a mechanical one and not an electronically depicted one.
Put someone on the localizer and then tell them to spin the OBS and observe the needle. It'll rotate angularly around but the center of the needle will stay centered in the middle of the instrument right on the center dot.
Because all they're doing is rotating the ILS Instrument around in circles inside their DG. Literally on a mechanical one.
If they were flying an ILS with glideslope, both needles just spin around.
The "plus" shape would just rotate around as they spin the OBS.
The needles that make up the plus don't care at all about the OBS being rotated when flying a localizer and a glideslope. They just "care" about showing where the aircraft is in relation to the two radio "beams".
The drawings and electronic versions make this relationship of how original HSIs were created hard to see. It's just one Instrument stuffed in the center of another Instrument.
Helpful?