Could dark matter be "inside-out" reality? Where we are, and in the conventional galaxies etc. that we see, there is a whole lot of relatively uniform space but in isolated places within that space, there are pockets of concentrated "mass" which is highly organised into all the objects we observe and study, and out of which we are made.

Now suppose that there were regions of space where that is all turned on its head, and there were huge vast swathes of relatively uniform matter, and within that matter there are pockets of "space" which is highly organised into some other form of object, out of which other objects and beings are made.

Obviously on the first, and simplest test - the question of "how much" dark matter there is, this theory passes the empirical test of reality with flying colours, i.e. astronomers seem to observe much more mass in the regions of hypothesised dark matter in accordance with what this conjecture implies.

What other testable predictions does this conjecture make, and how well do they fit our observations?
As a disclaimer: This answer does not include references, and is mostly based on my first intuition when reading the question. I can see that it can potentially be an interesting question to discuss.

First, we live in a solar system, in a galaxy, in a galaxy cluster. That in itself means that there is a lot of matter around. We are also (at least very likely) living in an overdensity in the universe, meaning that there is extremely much matter where we are compared to the average in the universe. 

If we assume that we do live in an underdensity of the universe, that would mean that there is more matter present in the universe on average compared to where we are. However, if pockets of space in a distribution of regular matter was dark matter, then it sounds like there would be a larger amount of regular matter than dark matter in the universe, something which does not agree with our estimates. 

If we assume for a minute that it is the other way around (that we live in an area with a lot of matter, not a lot of space), then this theory will struggle with explaining some of the things that point towards the existence of dark matter. One such thing is the rotation curves of galaxies, which indicate that there is more mass presence than there is seen. I expect that there would not be enough space within a galaxy to create such "pockets" of space. Say, however, that we assume this was possible, then the voids we observe between galaxies and galaxy clusters would be even more gigantic pockets of space, which would mean that there would be an enormous amount of dark matter, which I would think would be inconsistent with what we see from the CMB.