Why shield a balanced cable?
Here's one reason for it, I think.

Inductance is, among other things, a function of cross-sectional area and number of turns. If you run two wires in parallel then this cross section is equal to the distance between the wires × the length of the wires. The number of turns is equal to 1. 

In the world there's a lot of magnetic fields swooshing back and forth, one of them being in your house with the mains power. So you got an active magnetic field around you, constantly switching, constantly being picked up by your two parallel wires.

So what you essentially got is a weak air core transformer, the mains... N = 1? Or something.. It's messy. And your straight wires, N = 1. Both have some inductance above 0. 

By twisting the wires the inductance of the two wires reduces considerably, imagine a figure 8. It cancels out the current because whatever magnetic field that is penetrating the upper hole is also penetrating the bottom hole and thus averaging out to zero. But the figure 8 is just a half turn, if you make as many turns as you can then the noise is attenuated greatly. 

Now, to the shielding. What if... You could.. make use of eddy currents. This means that as magnetic fields are changing in the material, they will induce a current, but the shielding is taking a huge part of the energy by allowing most of the the current to circulate in the shield. And whatever the sign of the noise ends up being, it will be sucked out from the shield because it's grounded.