If you accept the notion that all disc galaxies are oriented on exactly the same plane (which is current scientific consensus I believe), then it doesn't seem that far-fetched to think that they all rotate the same way as well.
They're not all aligned on the same plane, they're somewhat aligned along the matter filaments and sheets (including the suspected dark matter) that make up the large scale structure of the universe.
"Galaxies are not distributed randomly in the cosmic web but are instead arranged in filaments and sheets surrounding cosmic voids.
[...snip...]
We found evidence that the spin axes of bright spiral galaxies have a weak tendency to be aligned parallel to filaments. For elliptical/S0 galaxies, we have a statistically significant result that their spin axes are aligned preferentially perpendicular to the host filaments;..."
"Galaxy shapes are not randomly oriented, rather they are statistically aligned in a way that can depend on formation environment, history and galaxy type."
I agree. It looks pretty random. Its probably even more random since we can't reliably eyeball the chirality in most of photographed galaxies so half the angles in one of the axis are practically mirrored at first glance.
(Imagine standing in a room with a bunch of CDs suspended horizontally from the ceiling. From any vantage point you can see some CDs edge-on, some from the top, some from the bottom, etc. But they're all oriented in the same direction.)
This situation is trivially detectable though. Simply plot the inclination angle of disc galaxies as a function of azimuth and inclination, if it's like you say, there will be a very clear banded structure.
I suspect that in your example you would observe a number of different apparent orientations, but that there would be some types of apparent orientation that could never occur.