Editing and anchoring a copy of a shape drawn with pencil tool

[Gully]

I am very inexperienced with GIMP and it seems like my problem is very basic, so I apologize if this is a trifling question. I am trying to draw a pair of bird wings. I used the Paths tool to draw my initial wing shape and it was fine. My plan was to take that shape, copy it, and use a few other tools to draw the opposite wing. I have attached an image of my starting shape and 2nd screenshot that sketches out the intent of my next step. From the history in that screenshot, you can see that I Selected the starting shape, used the Flip tool, and then moved the new shape into a position that would approximate the opposite wing. No matter if I used Paste (as various  methods) or anchor or what, I cannot for the life of me, figure out how to make the new shape appear in the layer as a permanent image/drawing/whatever. I've tried various methods with Paste, like As New Layer, and/or duplicating the layer, but when I anchor or merge down the layer, the new drawing disappears like it wasn't there to begin with, even though I have not changed the foreground color.

I have been at this particular task for 2 hours. I could have hand drawn the opposite wing (I could have hand drawn 10 pairs of wings) by now, but I am trying to learn a little about GIMP's tools and processes. It seems like a powerful software and knowing some basics seem useful. I thought I understood some basic concepts of GIMP, but I guess I don't. 

I apparently have no idea how to do what I want. What is the correct process for using a starting shape as a layer, editing it (like Flip or Rotate or Handle Transform), and pasting or anchoring or merging the new shape into the starting the layer?