
I put the apparatus back to how it should be, except that part A is the left side part, there is an identical part on the right.
Rather than being on the inside of the 'sheath' consisting of a left half (L) and a right half (R), the edge labelled A and the edge labelled B fit together in a tongue-and-groove setup.
So, rather than having a single edged saw these wasps have a double-edged saw (one edge pointing back (dorsal) and one edge pointing forward (ventral).
My interpretation is that the stiffer valves (R & L) are pushed into the twig. and then part A slides into the tiny hole formed by the tip of L; A and L then work back and forth to cut a slit in the twig.

NU11057