Eagle 9.5.x - custom flip-able layers
I recently learned that Eagle has a provision for this case.
The layers 53/54, 55/56, 57/58 and 59/60 are hidden by default. But can be used for custom flipable layers.
Other layers though, don't behave like this.
You can unhide them by using the command:

LAYER 53 customnametop;
LAYER 54 customnamebottom;