You can reorient the "up" direction and ground plane rotation at the same time you change the origin.
To update the orientation of the z axis, do one of the following and press Enter:
To maintain or change the orientation of the x axis, after setting the z axis, use one of the following methods:
Note: You will only be given the option to update the x axis if you either view the z axis by pressing Tab after clicking to make your selection, or change it through pressing Alt, before pressing Enter. If you press Enter directly after selecting the ground plane via Update Origin, the option to change the x axis will not automatically appear.