Test for symmetry? [Please]

(This was posted in a different sub-section of the forum, but has remained unanswered for over a month. I'm posting it here in the hopes of getting a respondent.)

I'm doing a psychophysical study on motion direction estimation and the direction-dependence of the estimation errors.
My current problem is determining whether the perceptual bias of a motion direction is symmetrical around a cardinal direction.

So what I have are:
  • multiple measurements of perceptual error (dependent variable, continuous), and
  • several test directions (independent variable)

Because the test directions can be paired about the axis of symmetry, I thought I could test for symmetry by determining that each pair of distributions came from the same population (like an unpaired t-test). However, I do not know how to do this for multiple pairs; ANOVA compares multiple conditions against each other, but I was told that using it to test FOR the null hypothesis is a no-no.

Additionally, the error distributions are not normal, so it would be better to find a non-parametric test.

Maybe there is a better way of testing for symmetry than paired conditions, but I only have a basic grasp of statistics.
Any help would be much appreciated.