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How do you invert a clipping mask, so that instead of the underlying layer appearing "inside" the mask, it appears outside it?

Lets say I have a star shape on top of an orange rectangle.

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When I select the star and rectangle and add layer mask, the orange rectangle shows through the star. I can move the rectangle around inside that mask by using Object > Clipping Mask > Edit Object.

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But I want the mask to cut a hole in the orange rectangle so that the blue background shows through, and the position of the orange rectangle can be moved or edited independantly of that mask.

This is what it would look like (which I achieved destructively):

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I can't get the 'mask' or 'invert' options to appear on the transparency panel. The shape is not / can not be compound.

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You don't really want a clipping mask here, or an "inverted" clipping mask. You want a compound shape. A clipping mask hides artwork outside the mask shape. You want to hide portions within a shape.

Easy ways to generate compound shapes...
(Which are different than Object > Compound Path)

Star above rectangle in stacking order, and select both.

  • Hold down the Option/Alt key and click Minus Front on the Pathfinder Panel.

    • By default Pathfinder operations auto-expand (since CS4), making them destructive. If you hold down the Option/Alt when you click a Pathfinder Button, it tells the operation to not auto-expand, resulting in a compound shape. (This is the reverse of how Pathfinder operates prior to CS4.)

  • One can also group the shapes, then use Effect > Pathfinder > Subtract to get the same, non-destructive, result.

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The difference between these two method is primarily how they appear in the Layers Panel. The Group with the Effect, will be seen as a standard "Group" in the Layers Panel - and the Effect appears in the Appearance Panel when that Group is selected. The Compound Shape generated by the Pathfinder Button will be listed as a "Compound Shape" in the Layers panel. Other than the name it will appear similar to a Group, but there is no effect for it when looking at the Appearance Panel.

Using Object > Expand Appearance on either method results in destructively "baking in" the compound appearance.

The Pathfinder commands create compound shapes which are different than Object > Compound Path > Make. I can't definitively state the difference between the menu's compound "paths" and pathfinder's compound "shapes" - I simply know they operate differently (and are labeled different in the Layers Panel.) I do not believe there is any way for a user to intentionally create a compound shape, other than the option/pathfinder operation detailed above. All user-driven menu commands create compound paths, not shapes.

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