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Dama

Alternatively, why does the matter that we see only rotate in one direction?

That's the better way to phrase the question. Galaxies start off with everything (visible and dark matter alike) going in all directions. Visible matter experiences friction with other visible matter; the upshot is (and, sorry, this is by no means obvious, but it does work) that the matter segregates over time into a low-angular-momentum, flat, corotating disk and some high-angular-momentum ejecta. Dark matter, absent friction, never gets modified this way. The Earth happens to be in a galaxy where this has taken place, and we happen to be in the disk.
 
My thanks to ben m and Dancing David. It's much clearer now.

I certainly didn't realise that the residual angular velocity of visible matter was because of ejection of high angular velocity components, I sort of thought that opposing angular velocities cancelled each other out and left what we see now.
 

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