It was clear that the structure of the box would derive from the two "wings" that fold in being able to connect to the body of the box in a stable way.
Buttonholes and a button seemed like an obvious choice. But in order to make it reversible, I couldn't do a single buttonhole on each side, through both layers of each folded "wing" - that would only allow the box to assemble in one direction. Eureka! Make a pair of buttonholes in each wing, lined up to fall on top of each other perfectly so that the box could be buttoned up through all the layers!
I started out by marking the buttonhole positions based on the pinned-together box...but only on ONE side of each wing.
I then stitched and cut open those buttonholes, folded each wing into its finished position, and pinned through the finished buttonhole into the layer below, giving me a precise position that would line up with the first buttonhole. (Apologies for the blurry shot!)
With all the buttonholes completed and cut, the wings can be folded back into place, and a pin through all the openings will give you your final button position.
Making it reversible means you will have to sew two buttons at this position - one on either side of the fabric. I recommend using a button with a long shank - they are easier to work with back-to-back than a flat disc button, and the long shank is also friendlier when it comes to buttoning through four layers of fabric. I happened to have four old wood buttons with the look of bamboo - perfect for the theme!
So, we know it works...but, is it reversible?
Worked like a charm! Though a quick pressing is probably advisable before reversing, just for tidiness.
A traditional frog closure will be a perfect finish for the red-and-black side.
I'm not sure yet how I'll finish off the blue floral side. Maybe a fancy button with a thread chain closure in some silk buttonhole twist. And, for a strap? Simple - make one out of each fabric (+ interfacing for stability) in the desired length, just as you would a tie belt or a dress strap, topstitch the edges, add a buttonhole on each end, and button the strap right to the box! Alternatively, you could use a length of jewelry chain - something about the weight you'd use for a charm bracelet - add a lobster-claw clasp to each end, and clip it right to the button shank.
Now all you need are some fabric fortune cookies to fill it with, and you're good to go!