Is there a shortcut for cycling through these viewport options? I've been looking in the hotkey manager, but nothing related to this functions comes up.

The ‘y’ key will toggle between Hide and either Show All or Ghost depending on which one you have set.

ah, thanks, It's called ‘Cycle Handle Mode’ in hotkey manager. I thought that was to do with ‘handles’ but that might be H8 ghosting in my mind.

The two relevant hotkeys are called “Cycle View Context” and “Toggle Local/World View Context” and can be found under the /Houdini/Panes/Geometry Viewers/Operations/View Operation context. You're looking at the wrong key.