What is meant by "the symmetry of an interaction"? My understanding of a symmetry is this: apply an operation (e. g. parity inversion) to a system. If it behaves the same afterwards, it is symmetric under that operation.
Now, quite often I see statements like this:

Isospin is regarded as a symmetry of the strong interaction under the
  action of the Lie group SU(2), the two states being the up flavour and
  down flavour. [...] In simple terms, [the] energy operator for the strong interaction gives the same result when an up quark and an otherwise identical down quark are swapped around. 

(from https://de.wikipedia.org/wiki/Isospin)


*

*How can the strong interaction "have a symmetry"? An interaction is not a one-time operation like parity inversion. Is the meaning of this that any strong interacting process does not affect the isospin? Or that reversal of all isospins in a system does not change the behavior of the strong interaction?

*I also don't see how in the specific example from above a down quark is suddenly "otherwise identical" to an up quark except for its isospin. Wouldn't up and down quarks always differ by mass and electric charge?

 A: A general way of understanding what a symmetry means for physicists is to think of an operation that generates new solutions to the equations of motion from previous known solutions. In classical mechanics, for example, if you take a two-body problem in which the potential energy that governs the interaction between the two particles is central (only depending on the distance between them), you can take a known solution (for instance, the one in which the center of mass of the system lies on the origin of your coordinate frame) and translate it by a constant distance, generating a new solution (a solution in which the center of mass is not in the origin of your coordinate system). If you had an interaction which depended on the absolute value of the position of those particles with respect to your coordinate system, however, translation would not, in general, take the system to a new possible solution; the evolution of the system would be essentially different.
That intuition is readily applicable to field theory, where the role of the equations of motion is played by the field equations (Maxwell's equation in the case of electromagnetism, or Yang-Mills' equations in the case of quantum chromodynamics). So what is meant by "a symmetry of the interaction" is that if you have a field configuration that solves your equation of motion, and you swap the flavours of the particles involved, you still get a solution to the equation of motion. 
