Oops, I actually have solved the issue. When I first got the printer I found that one of my 8mm Y shafts had about half a mm of play axially which I had fixed using an endcap with a nylon screw. This nylon screw wore down from friction without me realizing and the play came back. Interestingly, the inertia from print moves was not enough to shift the rod, however travel moves did hence the consistency (always start of a new layer, right after a large travel move).
Have you checked if your short belts are tight enough? If there is slack in them they could allow for some misallignment.
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