In essence my question can be stated as follows: fill in the analogy

$$ \text{cup product} \qquad\qquad \leftrightarrow \qquad \text{Samelson product} $$

$$ \updownarrow \qquad\qquad \qquad\qquad \qquad\qquad\updownarrow $$

$$
\quad \quad \text{cup}_i \text{-product} \quad \qquad \leftrightarrow \qquad \qquad \quad\qquad\text{?}\qquad \qquad\quad\quad
$$
It is known that Samelson products (for a loop space) are Hilton-Eckmann dual to
cup products (see e.g., Arkowitz, Martin: Commutators and cup products. *Illinois J. Math.* **8** 1964 571ā581. )

Taking this a bit further, the construction of the Steenrod algebra uses the fact
that the reduced diagonal $X \to X\wedge X$ is $\Bbb Z/2$-equivariant.

It is not hard to show that the Samelson product also has an equivariance. This suggests
to me that the graded Lie algebra structure on the homotopy groups of a space can be refined to take this into account.

Any ideas?