For some reason the pivot point on my object has become rotated (other objects show normally oriented pivot points). I have to set the "transformation orientation" (see the bottom of my screenshot) to be able to see it though. Does anyone know how to change it back? That is, change it to be aligned with the "world orientation".
This is intended behavior, if you are using Local Coordinates the transform gizmo orients itself to match the object local coordinates.
If the object is rotated so is the pivot, to match the objects orientation.
Either clear the object's rotation if it is not wanted (Alt+R), apply it so it becomes part of the object geometry (Ctrl+A), or use Global Orientation instead.