  Random operators constitute fundamental building blocks of models of complex
systems yet are far from fully understood. Here, we explain an asymmetry
emerging upon repeating identical isotropic (uniformly random) operations.
Specifically, in two dimensions, repeating an isotropic rotation twice maps a
given point on the two-dimensional unit sphere (the unit circle) uniformly at
random to any point on the unit sphere, reflecting a statistical symmetry as
expected. In contrast, in three and higher dimensions, a point is mapped more
often closer to the original point than a uniform distribution predicts.
Curiously, in the limit of the dimension $d \rightarrow \infty$, a symmetric
distribution is approached again. We intuitively explain the emergence of this
asymmetry and why it disappears in higher dimensions by disentangling isotropic
rotations into a sequence of partial actions. The asymmetry emerges in two
qualitatively different forms and for a wide range of general random operations
relevant in complex systems modeling, including repeated continuous and
discrete rotations, roto-reflections and general orthogonal transformations
