Two distinct types of symmetry (or of viewing symmetry) can be distinguished: object symmetry and matching symmetry. The two types differ in how the landmark configurations relate to the axis or plane of symmetry.
Matching symmetry (A in the diagram below) is the symmetry of paired structures that are present as separate, mirror-image copies on the left and rigt side of the body. Their symmetry can be studied by matching the pairs together. Fly wings or human hands are good examples of matching symmetry.
Object symmetry (B in the diagram below) is the symmetry of objects that are symmetric in themselves. This means that the axis or plane of symmetry passes through the landmark configuration. This means that some landmarks may lie in the midline or mid-plane, while others exist as corresponding pairs on the left and right sides. A typical example of object symmetry is the head of humans.
From Klingenberg et al. (2002)
For morphometric analyses, it is important to distinguish these two types of symmetry. The analysis of landmark configurations with matching symmetry is similar in many respects to the analysis of configurations with no symmetry at all, except for the need to keep track of the two configurations for each individual. In contrast, object symmetry requires a special kind of analysis.
Analyses of object symmetry in MorphoJ follow the recommendations of Mardia et al (2000) and Klingenberg et al. (2002). These analyses produce separate components of symmetric variation and asymmetry. Further information and detailed explanations were presented by Klingenberg et al. (2002) and a recent overview can be found in Klingenberg (2015).
Klingenberg, C. P. 2015. Analyzing fluctuating asymmetry with geometric morphometrics: concepts, methods, and applications. Symmetry 7, 843–934.
Klingenberg, C. P., Barluenga, M. & Meyer, A. 2002 Shape analysis of symmetric structures: quantifying variation among individuals and asymmetry. Evolution 56, 1909–1920.
Mardia, K. V., Bookstein, F. L. & Moreton, I. J. 2000 Statistical assessment of bilateral symmetry of shapes. Biometrika 87, 285–300.