North Arrows and True North
When you insert a north arrow onto a page in ArcMap, by default it will align to north of the data frame, the page in other words; also called “grid north”. The coordinate system is not taken into account.
In its default setting, the north arrow will rotate with the “rotate data frame” tool. However changing the coordinate system from UTM Zone 8 to 10 has no effect at all on the arrow’s angle (the change from zone 8 to 10 is big).
As of ArcGIS 10.1 (circa 2013) there is a property to change the orientation to follow True North. In this mode the Rotate Data Frame tool has no effect on the arrow position, while changing the coordinate system does change the arrow’s direction.
In this figure each map sports two north arrows, one aligned to Grid north and the other to True north: