Let $X$ be a pointed topological space and let $\tau:\Sigma^2X\to\Sigma^2X$ be the map swapping the two suspensions. I want to understand the map on homotopy groups induced by $\tau$.
I tried computing some simple examples. For $X=S^0$, $\tau$ is just a reflection, so on $\pi_2$, $\tau$ is mulplication by $-1$. For $\pi_3$, when we identify $S^2\simeq \mathbb C P^1$, $\tau$ sends $[z:w]$ to $[\bar z:\bar w]$ (at least up to homotopy). Composing with $\tau$ amounts to flipping two coordinates on the Hopf map and so doesn't change the homotopy class.
On the other hand, if $\alpha\in \pi_n(\Sigma^2 X)$ is of the form $\alpha=\Sigma^2 \beta$ for some $\beta\in \pi_{n-2}(X)$, then $\tau\alpha$ is the same as precomposing $\alpha$ with the swap map on $S^n\simeq \Sigma^2S^{n-2}$, which has degree $-1$, and so $\tau\alpha=-\alpha$.
Is there a general description for the induced map on $\pi_n$?