What is symmetrical output voltage swing?
The voltage swing is a term that describes how much the output of an OpAmp can swing around its equilibrium state. It is also measure of how close the output of an OpAmp can move to either one the supply rails.

If the output can move the same amount to higher and lower voltages the swing is said to be symmetrical.

E.g. for a +10/-10V supply the swing could be from -9 to 9V, with 0V at the output when the signal is zero. Another possibility would be to have a +10/0V supply and a swing around 5V, for example from 1V to 9V.