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To perceive a symmetric pattern, an observer needs to find correspondence between two image elements across the symmetric axis, implying an excitatory relationship between perceptual mechanisms responding to these elements. To perceive a 3D structure in a random dot stereogram (RDS), the perceptual mechanisms tuned to different disparities would inhibit each other. We investigated whether putting corresponding elements of a symmetric pattern in different depths would affect symmetry detection. The symmetry patterns consisted of dots (0.19degx0.19deg) occupying .5% of the display. We measured the coherence threshold for detecting symmetric patterns rendered on 14 possible 3D structures that were produced by an RDS. The coherence threshold for symmetric patterns on a slant surface was similar to that on a frontoparallel plane even though in the former the depths of the two sides of the symmetric axis were different. The threshold increased dramatically when one side of the axis inclined toward the observer while the other side inclined away though the depth difference between the two sides was the same as that in the slant condition. The threshold reduced on a hinge configuration whose joint coincide with the symmetry axis. Our result suggests that co-planarity is a decisive factor for symmetry detection.