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It's hard to tell without seeing exactly what you're doing, but the start and end points of the selected section should only flip if you try to put the end point before the start point, so maybe you selected the part the goes in the other direction. I'd recommend using the Shift Route tool to move the overlapping sections apart (which it sounds like you may have already done), then drag the position slider above the map to double check that you are on the correct section when selecting it.
I'm trying to re-plot a section where the out and back tracks overlap. I select the first (aka start point) then I select the second (aka end point) of the section. When I select the second point the start and end flip positions. Why?
I shfted the section and I carefully selected the points on segments going the correct direction and again the start and end flipped. Why?