welll your right in a sense that the system would start out 3D, over time however, because of the rotational axis/heavier objects concept, things start to become uniform..
another example to further my explanation is the centrifuge(unless you know some stuff about dynamical systems/chaotic theory) which has to do with matter-heavier matter moves to the bottom of the glass while lighter matter moves to the top.
Now in our case if you think of a cylinder spinnin on its middle axis
the matter starts to collect in this 3D rotational way like the dog spraying water but now in terms of matter and the centrifuge, heavier stuff starts to move to the normal middle plane of the axis(by "gravitational pull") like you see the centrifuge matter move to the bottom but in this case the middle because on earth you'd have to have a tube that runs from the southpole to north poel to see this effect.
And as heavier stuff moves closer it pulls in other sorts of matter.
so in essence you got 3 things working here,
(1) cylindrical rotation-something heavy in the middle creates
this effect.
(2) planar middle rotation-the centrifuge idea that heavier
stuff tends to move towards the middle normal plane.
(3) matter at the poles- you still can get local systems rotating near the poles but because of the centrifuge idea they still have attraction tot he middle.
2 things to note
A)This is over a large amount of time so is very slow.
B)The get a better idea of these effects you need to know some
general physics/mechanics concepts of conservation.