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Before I do a club demo, I like to run through the process as refresher practice. I felt I might as well make and share my practice in this video.

I am sure that a cross grain box may be turned several ways. This way is my favorite using closed chuck jaws for the initial mount instead of a screw chuck. I like it because I do not have to be careful to not drill too deeply.

The process does require at least three mounts each for the base and the lid.
Base:

1. Press against closed jaws while turning a sacrificial mortise on what will be the inside;
2. Mount with sacrificial mortise; Turn underside including a mortise; complete with final finish.
3. Mount with mortise on underside; Turn remainder of interior and exterior. Finish

Lid:

1. Press against closed jaws while turning a sacrificial mortise on what will be the underside;
2. Mount with sacrificial mortise; cut a shallow sacrificial tenon on what will be the top in wood that will be cut away;
3. Mount with sacrificial tenon; Shape underside of lid; fit to base; cut another shallow mortise; Finish underside of lid;
4. Mount with shallow mortise on underside of lid; turn top of lid.

In this case I used two different woods for lid and base. The lid is zebra wood with its striking grain. The base is white oak which is pretty but more subdued than the zebra wood.

This box is finished with beeswax and mineral oil which was also used as sanding media.

Enjoy.