I have the following object:
How do I modify the origin so that the origin has the X axis normal to the face shown, but the object itself remains in the same position in space?
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Because you've applied the rotation to your model, the easiest possible way to do this is rotating it manually.
Turn on the snapping tool by using the magnet icon in the bottom toolbar and set the snap element to Increment.
Enter top view by pressing Numpad 7, then enter Orthographic Mode by pressing Numpad 5. Use the keys G,Z to rotate along the Z axis and hold Shift until the flat faces facing the x axis are exact straight lines.
While still in object mode, use Ctrl+A and click Rotation to reset the rotation to 0,0,0.