Cannot edit UV mesh? It seems I cannot find a way to edit my UV meshes.

I selected an object in object mode, went to edit mode, selected part of the surface, did a Project from View - and the change was reflected immediately in rendered view. I rotated my object , selected different vertices and projected again - so now my image shows up in two places on my object, as desired. 

However, I wish to fine-tune everything, i.e, scale, shear, rotate, translate the UV meshes of the two islands I just created. This should be possible in the UV/Image editor.
With my object still in edit mode, I can pick the UV map to be manipulated and the image to be used. In paint mode, I can see the UV meshes, but there is no way to select and manipulate it. In view mode I can do some selecting (All, Circle, but border select does not work at all) somewhat: What I select in the UV editor (with "Keep selection in sync")  reflects in the selection in the 3D view - but I cannot see what I selected in the UV view, nor can I transform the UV mesh selected (i.e., doing some "G" or "S" or "R" blindly(!) does not change the final rendered apparence). Without "keep selection in sync", no selection seems to be possible at all ...

I wish I could reach a situation like this, where the yellow lines and thick vertices in the UV editor window indicate that things can be transformed:


But, alas, all I can achieve (and I have meanwhile crawled back and forth through all menus for over an hour) is either white lines in paint mode or no lines at all in view mode ...

What is it that I keep doing wrong (Blender 2.79)?
While in Edit mode in 3D View, and in View mode in UV/Image Editor, the View menu in UV/Image Editor has an item UV Local View with a checkbox. (With other views in either the 3D or UV window, the menu item may not be present).

Uncheck it.