Summary: We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system \[ \begin{cases} -\mathrm{div}[F^\prime(|x|,|\nabla u|^2) \nabla u] = F^\prime(|x|,|\nabla u|^2) |\nabla u|^2 u \quad & \text{in } \mathbb{X}^n, \\ |u| = 1 & \text{in } \mathbb{X}^n ,\\ u = \varphi & \text{on } \partial \mathbb{X}^n.\end{cases} \] Here \(\varphi \in \mathscr{C}^\infty(\partial \mathbb{X}^n, \mathbb{S}^{n-1})\) is a suitable boundary map, \(F^\prime\) is the derivative of \(F\) with respect to the second argument, \(u \in W^{1,p}(\mathbb{X}^n, \mathbb{S}^{n-1})\) for a fixed \(1<p<\infty\) and \(\mathbb{X}^n=\{x \in \mathbb{R}^n : a<|x|<b\}\) is a generalised annulus. Of particular interest are spherical twists and whirls, where following [M. S. Shahrokhi-Dehkordi and A. Taheri, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26, No. 5, 1897–1924 (2009; Zbl 1172.74021)], a spherical twist refers to a rotationally symmetric map of the form \(u:x \mapsto \mathrm{Q}(|x|)x|x|^{-1}\) with \(\mathrm{Q}\) some suitable path in \(\mathscr{C}([a, b], \mathrm{SO}(n))\) and a whirl has a similar but more complex structure with only \(2\)-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions.