The E and H maps in the EHP sequence have models that aren't too hard to define. To review, the E map is induced on homotopy by $E: \Omega^n S^n \to \Omega^{n+1} S^{n+1}$ sending a map to its suspension. The H map has a nice model once one knows about James's theorem that $\Sigma \Omega \Sigma X$ splits as $\bigvee_i \Sigma X^{\wedge i},$ where $X^{\wedge i}$ is $X$ smashed with itself $i$ times. When $X$ is $S^n$ one has $\Sigma \Omega S^{n+1}$ then projects onto the "quadratic" wedge factor $\Sigma (S^n)^{\wedge 2} \cong S^{2n + 1}$. The adjoint of this map is $H : \Omega S^{n+1} \to \Omega S^{2n + 1}$.

My question is "what about $P$"? That is, what is a model for this map? As I understand it, the usual approach is to show that $E$ and $H$ fit into a fiber sequence (when localized at two, or integrally through a range(?)) and then $P$ is the connecting map. I was disappointed to learn that despite its name $P$ only coincides with a Whitehead product in some cases. Of course, there are general nonsense constructions of connecting maps for fibrations in terms of the maps one already has in hand, but I want to know:

Is there an "intrinsic" model for $P$, one which doesn't rely on the $E$ and $H$ maps in its definition?

I would also be interested in an intrinsic model for the map on homotopy, say in terms of framed bordism.