A chain hanging from its ends forms a shape called a "catenary". This shape is perfect for a self-supporting arch, because the bending moment is zero at each joint. (If there was any bending moment on any joint to start with, it would have changed shape until there was no more any bending tendency). So even when you turn it upside down, the bending moment on each joint is still zero - so it doesn't try to change its shape, and it remains pretty stable.
After turning it upside down, any other shape will start to bend under gravity, and will tend to collapse much more easily.
