The fact that $\pi_1(S^1) = \mathbb Z$ is geometrically intuitive. It is linked to a wealth of concepts and results, notably the winding number, residue theorem on $\mathbb C$, etc. There is a combinatorics analog with Sperner lemma, when extended to labellings of the first triangle that wind more than $1$ time.
The fact that $\pi_n(S^n) = \mathbb Z$ still appeals to intuition, although more difficult to "visualize" for $n \ge 3$, and this kind of intuition is potentially misleading of course. The continuous map degree, fixed-point index, engulfing number, extend the idea of winding number. Sperner lemma has extensions to higher dimensions (although I am not sure about the extension of the multiplicity result).
Then, what happens with other non-trivial homotopy groups of spheres?
Take $\pi_4(S^2) = \mathbb Z_2$ for example. We could hope to have a boolean associated to each mapping (some kind of "orientation"?). I have not seen anything like that, although that may just mean I am too new to these topics. Similarly, there does not seem to be any combinatorics result associated.
Question: I kindly ask for references that present some cases of $\pi_k(S^n), k > n$, with either a geometric intuition, or a named concept (e.g. an "orientation" when $\pi_k(S^n) = \mathbb Z_2$) having interesting consequences, or a correspondence with combinatorics results.