John,  It just so happens that I bought the material you asked about
earlier this week.  It's called vulcanized fiber.  Thin sheets are
sometimes called 'fish paper', which is used as electrical insulation.

They make it by 'steaming' pieces of paper in zinc chloride.  This
makes the fibers somewhat sticky.  Then they stack multiple sheets
of sticky paper (depending on the thickness required) and press the
'blanket' of paper to form a solid sheet of the required thickness.

There is no glue, just paper fiber -- usually cotton fiber.  I use it
to replace the stiffeners in the bellows of barrel organs. It appears
to be an exact match to what was used originally.

There are several companies that make it and it comes in thicknesses
from about 0.002" up to over an inch (in 0.002" increments).  And it
comes in several colors.  One company that will sell it is Bar-Plate
in Connecticut.  Their web site is

  https://www.barplate.com/counterboard/vulcanized-fiber.aspx 

You can order it on-line and they sell it in small quantities by the
sheet (24" X 36") for about $10, depending on the thickness.  I use
0.020" thickness for small organs and 0.030" for larger ones.  It's
just like the material that I remove to rebuild the bellows.  When you
shear it, the raw edge looks just like the edges of the die-cut parts
you pictures and those that I've seen in the past.

Regards,
Craig Smith