There are several terms for this. Some call it pillowcase or envelope style. I thin the birthing name comes from Eleanor Burns. Birthing is a simple way to sandwich a small quilt. I have done it with a twin. I did not like the results. I regularly do 50x70" tops this way.
Generally, in making the quilt sandwich, you layer back/batting/top. For birthing, you do batting, backing, right side up then top, right side down. You smooth it as you would for a regular sandwich. Then you pin baste only the edges. You sew around the edges, leaving a gap through which you will turn it right side out. My next step is to play tug of war, at several angles, with the quilt between DH and myself, to flatten my sandwich again. Then you close the opening you left, pin baste and tie or quilt.
Friendly fish.
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