I think a burn test might help - poly usually frequently burns with black smoke and leave a hard or crispy ash/residue.
The residue also stays hot longer than ash from cotton or wool.
When I do a burn test, I light a candle, then cut a very small sample (about 1/8 x 1 inch) of the fabric, hold it with a tweezers, and light it on fire (have a fire/heat resistant ash tray handy - I also have a glass of water on standby).
If it contains a man-made fiber such as nylon, dacron, or polyester, it usually burns with black smoke and leaves the hard/crispy residue.
I can't tell rayon from cotton - there probably is a test for that, but I don't know what it is.
Wool usually smells like burnt feathers.
I recommend doing these tests outside. I've stunk up the house when I did them inside.