How do centrioles auto-locate to opposite sides of cell during mitosis?
During cell-division centrioles move towards opposite 'poles' and finally get attached to cell-membrane. Different factors influence &amp; drives this process, most still not discovered. So this mechanism is largely unknown. From the scientific literatures published on this  matter till recently we can say that this process should include: 


A structure along which the centrioles can migrate, 
Molecules associated with the movement of centrioles, 
Signal molecules triggering this process, 
Signal molecules determining the direction of migration. 


To answer 1., Microtubules form a intra &amp; inter-cellular network along which subcellular structures can move. Actin &amp; Myosin are also involved in some cells. For reference read Ganong's review of Medical Physiology. (the example you stated in your question about neurons is a substantially different process. But neurotransmitters are carried with the help of Microtubules. Chromosomal structures also are arranged with the help of them.) 

For points 2,3 &amp; 4 read this paper, these parts are very clearly described there. 

Thank you.