Different way to view action of fundamental group on higher homotopy groups

If G is a topological group, then the group acts on itself by conjugation, and this action is base-point-preserving. In particular, for an element $g \in \pi_0(G)$ and a higher homotopy element $\alpha \in \pi_{n-1} G = [S^n, G]$, one can check that the conjugate $g \alpha g^{-1}$ is well-defined and defines an action of $\pi_0(G)$ on $\pi_{n-1} G$. The space G is weakly equivalent to the loop space of the classifying space BG, and under this equivalence the conjugation action is taken to the action of $\pi_1 BG$ on $\pi_n BG$.

(Unfortunately, this doesn't work directly for the conjugation action of the loop space on itself because it is not strictly basepoint-preserving; one needs to use that there is a natural homotopy from a loop $\gamma * e *\gamma^{-1}$ to $e$ to produce the action.)

Any path-connected based space X is weakly equivalent to the classifying space of a simplicial group G; specifically, the Kan loop group of a weakly equivalent simplicial set. Even more, there is a Quillen equivalence between the homotopy theories of spaces and simplicial groups.

(Kan's original paper can be found here: http://www.jstor.org/pss/1970006)

I doubt this would be considered less geometric than the actions in your question, but if $(X,x_0)$ has a universal cover with covering map $p:(\tilde X, \tilde x_0)\rightarrow (X,X_0)$, then $p$ induces isomorphisms $p_*:\pi_n(\tilde X, \tilde x_0)\rightarrow \pi_n(X,x_0)$ and $\tilde p: D\rightarrow \pi_1(X,x_0)$ where $D$ is the group of deck transformations (covering transformations) of $(\tilde X,\tilde x_0)$.

Now for $\alpha\in \pi_1(X,x_0)$ and $\rho\in\pi_n(X,x_0)$ let $\tilde \alpha$ be the preimage of $\alpha$ under $\tilde p$ and $\tilde\rho$ be the preimage of $\rho$ under $p_*$.

Then $\sigma=(\tilde\alpha)_*(\tilde\rho)$ is in $\pi_n(\tilde X, \tilde x_0)$ and we get $\alpha\cdot \rho$ as $p_*(\sigma)$.

Now that I think about it, this is at least as geometric as the actions in your question, but I like the picture better. In the picture, the universal cover not only unrolls the elements of $\pi_1$, but it also unrolls the action of the elements of $\pi_1$ on the maps of spheres.

I prefer to do a more general case, which is useful anyway, and to use fibrations of groupoids. For spaces $X,Y$ define the track groupoid $\pi_1 Y ^X$ to have objects the maps $X \to Y$ and arrows $f \to g$ the homotopy classes rel end maps of homotopies $f \simeq g$, with the usual composition of homotopies. If $i :A \to X$ is a cofibration then $$i^*: \pi_1 Y^X \to \pi_1 Y^A $$ is a fibration of groupoids (give the "obvious definition", but it first appeared in a paper of mine in 1970, J. Algebra, although it is a specialisation of an earlier definition for categories, which has a different purpose). Now for a fibration of groupoids $p: E \to B$ there is an operation of $B$ on the disjoint union of $\pi_0$ of the fibres. Applied to the above case, this gives an operation of $\pi_1 Y^A$ on homotopy classes $X \to Y$ relative to maps $A \to Y$.

You can find this in Chapter 7 of my book Topology and groupoids (2006), with applications to, for example, a gluing theorem for homotopy equivalences. (and in essence in the first 1968 edition).

These ideas were found by thinking about maps $(S^n,x) \to (Y,y)$, and then generalising first replacing $S^n$ by $X$ and then the point $x$ by a subspace $A$ and forgetting $y \in Y$.

Sorry to be so long in giving an answer to this but till June, 2011, I was busy with another writing job!