Hmm - you did check the dish, right? The consistent difference between the sides makes me thing that perhaps the rim is not centered over the locknuts.

As for tension vs. alignment, once one has achieved sufficient tension such that the wheel is rigid and spokes are not going to loosen due to the rhythmic change in tension and road shock even tension is indeed less important. If that were not true a lot of wheels that have suffered some minor indignities would have to be considered unusable.

One other thought is that one could say that the goal of "truing" is actually to align the hub so that it is perfectly cent4ered and at a right angle at all points in respect to the rim.