Sounds like phenolic to me. Usually the reinforcement was paper but if you see weave, then cotton, nylon or glass may have been used. If you burn a small piece and it smells like hot brake pads (old type), then it again is likely phenolic. I would not breath the fumes from burning or the dust from machining the material. I would look up the safety data sheets for it and see what hazards it presents. Phenolic is a thermosetting resin and a good electrical and heat insulator, and withstands pretty high temperatures. A fire resistant type (FRPP) was ultimately developed before it was replaced by glass epoxy board (GRE or G10) for higher end printed wiring boards.