This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.
I know it's been proposed as a joke, but the idea that warp travel
requires a universal "up" isn't as silly as you might think. We know
that things in normal space affect subspace. What if the mass and
orientation of the entire galaxy, which is nothing to sneeze at,
affect subspace in such a way as to make travel more efficient if your
warp fields are generated parallel to the plane of the galaxy?
It's then more efficient for ships to align their warp drives with the
plane of the galaxy, so flying "up" and "down" in the galactic plane
(which is relatively thin, about 1/10th to 1/40th the diameter of the
galaxy) would take more energy. This also explains the banking into
turns and such.
If you have galactic-up and galactic-down to choose from, why always
the same way up? Probably a matter of protocol. Only "loser" races
don't adhere to the standard. You'll also note that many small ships
are vertically symmetrical, perhaps as their designers aren't quite up
to snuff when it comes to designing warp drives.
More support for this hypothesis: warp does really weird things at the
edge and at the center of the galaxy. The Great Barrier of TOS fame
("Where No Man Has Gone Before" [TOS] and "By Any Other Name" [TOS])
at the edge of the galaxy and the one near the center of the galaxy
(if you believe in Star Trek V) were each considered impassable and
gave the Enterprise a rough ride.
Kirk made a number of references in "By Any Other Name" [TOS] about
warp drive not working outside the galaxy, or something to that
effect. While the Kelvans of Andromeda had got that licked, this does
give the theory a little bit of support.
Another note: the subspace shockwave seen in Star Trek VI was both
planar and aligned with the direction of the Excelsior's vertical
axis, and shockwaves within the subspace rift of "Force of Nature"
[TNG] were also aligned coplanar to the ship.
Pete Carr points out the following bit of dialog from "Genesis" [TNG]:
Picard: Adjust the axial stabilizers [of the shuttle] to match the
attitude and rotation rate of the Enterprise.
A "universal up" would explain why this sort of thing doesn't happen
all the time; only drifting ships like the Enterprise need help.
 
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