Abstract
We identify a cyclic property of rotation sequences involving piecewise displacements β about arbitrary axes in three dimensions. Specifically, when transformation to the toggling frame is applied successively m times, for β=2π/m the original sequence returns. This main result unites several families of rotation sequences used for error-tuned control across quantum technologies, from NMR and MRI to atomic clocks and atom-scale computing. For the widest class of cycle, m=2, we highlight sequence duality where every narrowband π-element sequence has a broadband π-element counterpart, and vice-versa. Higher cycles (m>2) connect to polyhedral models for error-tolerant sequence design, characterized by vertex axes with m-fold rotational symmetry. We derive original sequences and outline their applications to spin control and spin decoupling.