I am seeing an effect in my prints that appears to be gear backlash, i.e. the print head does not go to exactly the same location when approaching from one direction as it does from the other.
The belts do not appear to be loose, nor do the pulleys or motors.
I looked in both Cura and Slic3r but did not find any adjustments for backlash.
Backlash is going to exist in any open loop system such as the Taz 6, how does one get rid of it?

Attached is an example photo. You can see the inside surface shifts for a while at the same height as the mounting hole that caused the direction of the head to be reverse of what is was before and after the hole.

What you are seeing there isn’t ne essarily backlash, though there is some of that present too. It’s more likely either a slicing or minor over / under extrusion or possibly retraction issue. What happens with layers where the hole is present is that you have a segment with several, starts, stops and active vector changes that aren’t present in the longer runs. Each time you stop and start or change vectors on a marlin based 3d printer you are basically telling the printer “go that way and accelerate at this speed until you stop” meaning the print head essentially moves faster in the middle and slower at the ends before it stops. The layers with the integral hole are shorter runs. The 3d printer firmware and slicing programs know about this non constant motion issue, so they compensate for it. Often newer firmwares and slicers use a linear advance variable to factor that in.

What happens when you see a difference like that is either your slicer is a tiny bit off, or you have probably a small amount of over extrusion that only manifests itself on longer runs. You can experiment with dropping your fillament diameter a bit or changing e steps and fling calibration, etc. But even the best 3d printers will show that from time to time.

Backlash also comes into play, but with the taz 6 it’s generally small enough you don’t notice it unless you go looking for it or add something like a stronger more supported boxed rail. It shows up as slight (like 0.01mm) offset every other layer usually, and is very hard to detect compared with extrusion inconsistencies between layers. You also run into things like gravity offset and sideways rod sway due to slightly off center belt pull mounting on most printers. There are multiple ways to combat that, some of which like the anti backlash z nuts for the taz leadscrews literally cost an arm and a leg.

What you describe, I would expect to appear only for a short distance (while the head is accelerating). The problem I am having extends the entire 5.75" of the inside of the back wall in the photograph. (you can’t see it in the thumbnail, you need to view the whole image).
The back wall is depressed about 0.5mm more at the height of the hole with the silver screw through it on the right wall, and above the end wall, than the rest of the wall.

Id look at the print preview layer by layer around the affected height. I’ve seen some really odd stuff previewing files before a print that i had to run through other apps to analyse and correct the problem.