I have been refolding my newest boxes with good quality paper.
I discovered a white and silver light weight card paper which folds fairly easily and does not feather or crack along the creases. It is a design from a K and Company product called simply "Foil". It is bound together with heavier scrap book type paper in a collection called "Wedding Paper Pad" by Marcella. The other papers, while gorgeous, are not suitable for folding. Although I searched for this paper on the Web I was not able to find it. Damn!
The cube-shaped twist box folded very nicely in this paper. The triangle on the handle is nicely centered because there is enough friction in the paper to keep it snug.
The paper worked well for the oblique walled box as well.
The next paper I tried was a lightly embossed iridescent paper which has a darker toned reverse side with no sheen. The sheet is slightly floppy and has a plastic feel to it so I was unsure whether it would hold a crease or have sufficient body to hold its shape. It came through on both counts. It helped to score if very firmly. The paper is marketed by the Paper Company in its Iridescent range and comes in American Letter size and 12" x 12". There are several different colors. You can buy it here: http://www.papercompany.com/metallicsEmbossed.htm
