Even if there was enough of an effect from galactic rotation to make a difference to stellar system rotation, there would be no "shift" in rotational direction from one side of the Galactic equator to the other.
The fact that storm systems rotate clockwise in one hemisphere and counter-clockwise in the other is just a matter of perspective. It is a kin to the fact the if you look down at the Earth from above the North pole you see it rotating counter-clockwise, but if you look down from above the South pole you will see it rotating clockwise.
Here's an animation to demonstrate. It starts by looking at the globe from the the plane of the equator (the black line), showing two storm systems in the Northern and Southern hemispheres. Note how they rotate in opposite directions as shown by the arrows.
The animation now rotates to a view looking down from the Northern hemisphere. The body of the globe is transparent show that we can see the Southern storm system as the body of the globe moves in front of it.
Notice that when viewed in line with each other, the two storm systems rotate in the same direction.
[PLAIN]http://home.earthlink.net/~parvey/sitebuildercontent/sitebuilderpictures/coriolis.gif
Now imagine that these storm systems are stellar systems in the galaxy. ( the globe would be greatly flattened) For such a system to move from one "hemisphere" of the galaxy to another, it would just move straight through the body of the galaxy. It wouldn't be constrained to following the outer "surface" of the galaxy. So looking down from the North of the galaxy, you would notice nothing different about the rotation of the system as it moved across the galactic equator. Similarly, you wouldn't see any difference in rotation direction of stellar systems on either side of the equator, even if they were effected by galactic rotation.