There is data on this question generated by research on the embodiment of spatial cognition. The idea here is that we mentally represent and construct space in relation to our bodies. From this perspective, the differences that you describe (left vs. right more complicated than up vs down) stem from the properties of how we perceive the world in our bodies:
Tversky (2008) describing research of Franklin and Tversky (1990):
We proposed that people keep track of the relative positions of the objects around them as they move by constructing a spatial-mental framework out of the three axes of the body and appending objects to it, updating it as the situation changes. We reasoned that accessibility of objects should reflect characteristics of the body axes and the world relevant to perception and action. The head-feet axis has salient asymmetries both perceptually and behaviorally; moreover, for the canonically upright observer, it correlates with the only asymmetric axis in the world, the up-down axis of gravity. The front-back axis separates the world that can be easily perceived and acted on from the world that cannot be easily perceived or acted on, but the left-right axis has few salient perceptual or behavioral asymmetries. This analysis predicts that, for the upright observer, things located along the head-feet axis should be fastest to retrieve, followed by things located on the front-back axis, followed by things located on the left-right axis.
Their data supports this reasoning. Furthermore in their experiments they also disentangled gravity from body orientation by having some participants recline:
For the reclining observer, no body axis correlates with gravity, so accessibility depends entirely on the body axes. In this case, things located along the front-back axis should be fastest because of the forward bias of perception and action.
This was the case as well.
Note that accessibility was operationalized as the speed in which participants could name the position of an object. Participants made very few errors in this task. So strictly speaking it does not answer your question about left-right errors. On the whole, this has nevertheless much overlap with your question. They could have increased the number of errors by imposing a short reaction time window, for example. In this case they would have expected these effects on the error rate.
Franklin, N., & Tversky, B. (1990). Searching imagined environments. Journal of Experimental Psychology: General, 119, 63–76. doi:10.1037/0096-34184.108.40.206
Tversky, B. (2008). Spatial cognition: Situated and embodied. In P. Robbins and M. Aydede (Eds). Cambridge handbook of situated cognition. Cambridge: Cambridge University Press.