I'm travelling at the moment and will be away from my development system for a few more days. When I get home, I'll check to see if the flipped vertices can be made permanent.
Having now explored the issue, I can confirm that flipped vertices are "remembered", so long as you don't change the line. But, as David has discovered, if you move, add or delete a vertex, ADE-GP "forgets" the line was flipped.
When you first create a line that uses a line texture (whether or not the texture is dimensioned), ADE straightens the line, horizontally or vertically as appropriate, recognizing the length of each leg, centers the straightened line of the selected texture and records the UV (i.e., location on the texture sheet) of each vertex. If you flip the line, ADE-GP flips the UVs, but there's no other record of the flip having taken place. If the file is compiled at that point, the texture "orientation" will be flipped as expected.
However, if you then move, add or delete a vertex, you almost certainly will have also changed the length of one or more legs. So, ADE-GP must create a new straightened line to recognize the new line leg length(s). (Otherwise, the displayed texture would be distorted.) In doing so, it creates a new set of UVs, replacing the old ones (which reflected the flip).
It would be very difficult to detect from a set of UVs whether or not the vertices had been flipped. But, when it does happen, it only takes a couple mouse actions on the part of the user to correct.
The way to avoid this difficulty is, as I described above, to envisage the way the line will be applied to the texture and draw the line with vertex 0 at the left or bottom end relative to the texture.
Don