There are informative and easily accessible images and videos that illustrate the Hopf fibration $S^3\to S^2$ by describing what happens to the fibers in the unit cube $(0,1)^3\approx S^3\backslash \ast$, e.g. those by Niles Johnson. Is there a similar way to visualize the map $S^3\to S^2\vee S^2$, which is the attaching map of the $4$-cell of $S^2\times S^2$ and allows one to the construct the first interesting higher whitehead product?

I understand that this map is not a fibration - it's the restriction of the product of the characteristic maps of the $2$-cells of $S^2\vee S^2$ but, if possible, I think it'd be interesting to be able to understand this visually in a $3$-cube as some kind of higher analogue of conjugation.

Note: this was originally a post on MSE that received no answers.