P: Is there a way to invert an Adjustment Brush mask?

You could tell us how. 

The gotcha to watch out for there, is that mask components (any of Select Sky or Select Subject, brushed or radial or graduated selection, colour range or tonal range selection) may have been combined using Add, Subtract or Intersect.

And in that case, inverting those selections alone will not achieve the opposite result. We'd also need to invert the operation which combines them, remembering that Intersecting two masks is a shortcut for Subtracting an inverted version of one from the other. Also, remembering that [A] Subtract [B] does not give the same output as [B] Subtract [A]. 

Consider [A] intersected with [B]: this is just their overlaps. [Not-A] intersected with [Not-B] will not give the opposite result. And there is no "extrasect" we can change the "intersect" TO. We can't just swap everything regardless.

To unpick, we might consider that [A] intersects [B] amounts to [A] Subtract [Not-B]. And a full inversion of THAT would be... what? [Not-A] Add [Not-B]... and not, as one might initially think, [Not-A] Add [B].

For another example: Not-A intersected with B, can be more simply conceived as [B] Subtract [A].

To invert that, we'd want [Not-B] Add [A] - or equally, which is the same thing, [A] Add [Not-B].

If all this seems logically tedious and abstract, and different from the normal relaxed and exploratory style of editing in Lightroom , well I would agree. This whole notion of inverting a mask is something of an intellectualisation IMO, and may not arise from what's actually needed to get the job done. Is this too rigid and preconceived an approach; what are the practical downsides of duplicating and inverting masks, e.g. what happens if something must later change - just some of my own reservations. The underlying selection tools are very welcome in themselves, though, no question.  

Please explain some more. A Luminance or Color mask?

If it is 'Luminance'- adjusting the Range Sliders can limit the mask to Highlights or Shadows. Is this what you mean?

My screen-clips are a monochrome image with the Red overlay showing the 'mask' on-

Highlights-

Shadows-

quote: "So after creating a selection of say a sky and adjusting you could work on the foreground Straight away."

But you already CAN do so not just straight away, but even beforehand, via your global adjustments.

FIRST you set the tonality and feel of the overall picture (accepting for the moment in doing so, that the sky or some other identified part may get pushed too bright or whatever in the process, but making a mental note to address that next).

THEN you address the sky, as necessary, assuming global adjustments can't manage that, via a local adjustment of some sort. In relation to the global processing.

You can tweak the globals further as needed, you can tweak the locals further as needed.

Throughout you are free to do more painting or erasing onto your local's mask, without then incurring the problem of how to match those same additions and removals, in reverse, onto a separate inverse mask. Because you'd definitely need to avoid either leaving any gaps between, or else making any overlaps of, those two. Sounds very inflexible to me.

The Keywords in your post are ..."in respect of those altered global settings". 

Balancing Local Adjustment against Global Adjustments (settings) is what we have to do for every mask.  Usually this works out find though it isn't alwasy practical especially when what is needed really is an invert of the existing mask.  For example, I need to drive the color balance of the sky very blue and reduce contrast.  Also though, I need to drive the color of the rest of the image very yellow with extreme contrast.   You can't always press the local adjustments far enough to offset the Global settings and get the intended result and in many cases the final result is mush.   The case to invert a mask has been made for years on this and many other forums.  It would be a huge time saver - or we use Photshop instead which has included mask invert forever.  

So, do the job twice.  Adobe needs to let us invert the current mask.