These pieces were used to protect linens for storage.
I don't know if they had a name.
Please enlighten me if you are aware of such.

This one is cotton, rather like a chintz, I think

although I have also had some that look more like barkcloth. They usually have cotton twill for tying.

I love the floral design ~
~ what some decorators are calling,
"shabby chic" these days.

I find these lurking in the bottoms of auction boxes of linens.
Sometimes they are still rolled.
Sometimes with the cardboard (card) still inside.
After ironing, the "housewife" would lay the linens on the pretty cotton,
roll and tie them into these lovely little packages.
This, to protect the linen fibers which tend to be brittle
and can break if stored folded.