Symmetry plays an important role in image understanding and recognition. In fact, symmetric patterns are common in nature and man-made objects and the detection of an image symmetry can be useful for designing effcient algorithms for object recognition, robotic manipulation, image animation, and image compression. This paper formulates the problem of assessing reflection and rotations symmetries of an image function observed under an additive noise. Rigorous nonparametric statistical tests are developed for testing image invariance under reflections or under rotations through rational angles, as well as joint invariance under both reflections and rotations.