Just like the X Mirror Editing on a mesh in Blender, it should be possible to enable X Mirror Editing on the same vertices in the UV Editor.
Currently the only ways to have symmetrical UVs is to, either, delete one half of the mesh and mirror it, with the 2 disadvantages that you, first, can’t rig it properly anymore, and second, the texture has to be placed either in the middle or you have to use the UV mirror offset setting in the mirror modifier, or, you have to select one vertex, mirror the selection with Shift+Ctrl+M while in Viewport mode and then go to the UV Editor and move the vertices along the Y-Axis with G and Y and if you want to move it along the X-Axis you have to press S and X to scale the 2 mirror-selected verices along the X-Axis.
It worked fine for me for several years now, but I think, having the same options for symmetrical editing avaiable in the UV editor just like in the Mesh Editor would make it much easier.