HAWK 37
You are correct that the flat card is effectively a symmetrical aerofoil, so its pitching moment about its AC is zero. But when dropped into the air it is not at its zero lift angle of attack (or at least is unlikely to remain at that angle) and it pitches about its C of G (not its AC). These two factors generate a leading edge up moment.
So it will usually tumble leading edge upwards (but occasionally screws up the classroom demo by doing the opposite).