It’s a very small lathe, and the new scale is intended to replace this one

which is a five year old lash up based on the inners of a set of very cheap callipers and some 3D printing. It’s getting tired and a bit cranky, and it reads sideways. However it does read positive values as the tool moves into the work, and negative if the tool moves out.
For size reasons I need to mount the head of the device behind the carriage (as the existing one) and have the cross slide move the slider.
Unfortunately, with the head mounted so that the display is correct, moving the cross slide into the work makes for negative readings while moving it away produces positive ones.
But I am sure I will get used to it.
With hindsight I could have bought one of the kind with a separate display, and simply turned the scale round, but that would have meant wires and more hassle to mount the display. So I will prersevere.