As far as I can tell (having recently come across an outstanding feature request on the official tracker), there is no way to maintain symmetry of a UV unwrapping that is comparable to the 3D mesh symmetry tools. And Copy Mirrored UV Coords appears to be broken (again, various reports floating about, and can't get it to do anything).
When you have a mesh that needs perfectly mirrored¹ UV coordinates, how do you achieve it? Lots of tedious manual coordinate editing? Hope to get lucky and have Blender's automatic unwrapping produce something that's already symmetrical? Or do you have a trick to make it easier?
(For bonus points, do you have any tricks for perfectly rotating an unwrapping that is symmetrical, but not aligned to an orthogonal axis?)
(¹ By "mirrored", I do not mean "duplicated", i.e. multiple faces that are mirror images in 3D sharing UV coordinates.)
Here's an example of the sorts of problems I run into on most of my models, using Sue:
In this case, I lucked out with the back of the head and the ears, but the face is rotated, and the eyes are both rotated and oddly distorted. (The ears appear symmetrical but are badly distorted, but that's my fault for not seaming them better. Don't worry about the ears. The point of this is just to show that Blender does not producing symmetrical unwrapping "out of the box".)
This is particularly a problem with models that are "not very round". Consider, for example, a humanoid that is seamed with the front and back as separate. Obviously, both of these should be symmetrical, but unless I use a UV that is much taller than it is wide (which I've found tends to result in very sub-optimal packings with lots of wasted space), I obviously can't just mirror everything across x=0.5.