The torque comes from angle of attack. See: Pitching Moment for an introduction to the physics. I'm afraid I don't know much about what causes the torque and how to calculate it. Perhaps another answerer can help, or you can ask a follow-up question.
If a symmetric wing was traveling at fixed airspeed (for instance, if it was fixed on a gimbal in a wind tunnel) it would cost just as much energy to continue turning the wing from edge-on around to the opposite angle of attack against the restoring torque. A little of the rotational kinetic energy is lost as heat via dissipative drag forces, so it will not rotate all the way to the full opposite angle. It would instead flip back and forth many times, each time at a slightly smaller angle, until eventually it stabilized in the lowest-drag configuration, edge-on to the wind.
If we take the wing out of the wind tunnel with its powered fan, and remove it from the gimbal, then there is a finite amount of mechanical energy available. The mechanical energy available to turn the wing comes its translational kinetic energy and its gravitational potential energy. Both heating up the air (drag) and spinning the wing (converting translational kinetic energy to rotational kinetic energy) reduce the translational kinetic energy and hence the speed. Of course, rising costs translational kinetic energy for gravitational potential energy, while falling transfers gravitational potential energy to translational kinetic energy.
Because the mass of the wing is very small compared to its surface area, it loses translational kinetic energy rapidly to dissipative drag forces, which means it slows down rapidly.
Because the wing is rapidly slowing down, the torque applied by a given angle of attack is rapidly decreasing. This means that it costs much less energy to keep turning the wing from edge-on around to the opposite angle of attack against the restoring torque than the energy that was available to start turning the wing.
The remaining rotational kinetic energy may be enough to rotate the wing all the way around past edge-on again, at which point torque is pointing in the direction of the rotation again, leading to cyclic motion.
If the wing is not symmetric (for instance, if it is curved), then it will also have one orientation for which a given translation velocity has more torque (potentially much more) than the opposite orientation, in which case it will accelerate angularly until an equilibrium is reached with drag. Since random pieces of flexible materials are not likely to be perfectly symmetric, this may be more important to your experiment than the wing slowing down. You could test this by dropping the wing at opposite orientations and seeing if one of them spins while the other doesn't.
Your statement that this does not apply to other materials is false. All that matters is having a low moment of inertia relative to its cross-sectional area, having the strip-like shape, being stiff enough to hold the shape, and either having a starting speed that is sufficiently higher compared to the terminal velocity in the starting configuration or having an asymmetric shape. You could cut a thin sliver of a metal sheet, use aluminum foil folded as necessary for stiffness, or cut a thin slice of styrofoam.
