We describe a discrete-time symmetry between three subsequent spinor locations \(X_{i},X_{i-1},X_{i-2}\) , where the tangential difference \(X_{i}-X_{i-2}\) is orthogonal to the intermediate point \(X_{i-1}\) located by reflection on the opposite side in normal direction. The binormal direction in the discrete Frenet-Serret frame is identified with a Hamiltonian vector function in a discrete-time Landau-Lifshitz-Gilbert (LLG) type equation describing spin and magnetism. The symmetry invariant \(C = X_{i}\cdot X_{i + lm \pm 1}\) with l loops in limit cycles of length m guarantees stable structures with differential shifts. The algebra provides for a highly nonlinear dynamics with pure imaginary octonion and quaternion spinor states \(\in \mathbb {R}^{3}\text { or}\in \mathbb {R}^{7}\) , where we find orbital wave states with characteristic quantum numbers, transition to chaos, hysteresis, and spontaneous helical emissions.