Edit: As Qiaochu remarks, the following is a special case of the Pontryagin-Thom construction. It's done wrong, though, because the framing of links is being ignored. If nobody beats me to it, I'll write an answer clarifying these points soon.
The Whitehead product $\pi_2(X,x_0)\times\pi_2(X,x_0)\to\pi_3(X,x_0)$ is induced by the attaching map ${\mathbb S}^3\to{\mathbb S}^2\vee{\mathbb S}^2$ of the $4$-cell in ${\mathbb S}^2\times{\mathbb S}^2$. Writing ${\mathbb S}^3=\partial{\mathbb D}^4$, ${\mathbb S}^2={\mathbb D}^2/\partial{\mathbb D}^2$, this can be described as $$\partial{\mathbb D}^4=\partial({\mathbb D}^2\times{\mathbb D}^2)=\partial{\mathbb D}^2\times{\mathbb D}^2\cup{\mathbb D}^2\times\partial{\mathbb D}^2\to {\mathbb D}^2/\partial{\mathbb D}^2\vee {\mathbb D}^2/\partial{\mathbb D}^2.$$
Now, the 4-cube / tesseract $\partial {\mathbb D}^4$ can be visualized as 
where the missing 8-th cube has the same vertices as the outer cube in the picture, but with disjoint interior. Alternatively, one might think of it as stretching out to infinity, filling the entire 3-space, plus the point at infinity.
In that picture, the map ${\mathbb S}^3\to{\mathbb S}^2\vee{\mathbb S}^2$ seems to be obtained as follows: Take the four 'middle' cubes horizontally circulating around the small inner cube. Together they form a solid torus ${\mathbb D}^2\times{\mathbb S}^1$ which can be projected onto ${\mathbb S}^2={\mathbb D}^2/\partial{\mathbb D}^2$. Similarly, looking at the remaining four cubes, stretching 'vertically' (including the outer 'infinite' cube), they constitute a solid torus, too, again mapping into ${\mathbb S}^2$. These two maps glue to give a map ${\mathbb S}^3\to{\mathbb S}^2\vee{\mathbb S}^2$.
The essence of this construction is the Hopf link immersed into ${\mathbb S}^3$, representing the two knotted tori. In fact, it seems that any embedding of a $k$-component link $L$ into ${\mathbb S}^3$ would give a map ${\mathbb S}^3\to {\mathbb S}^{2}\vee\ldots\vee{\mathbb S}^2$, with $k$ copies of ${\mathbb S}^2$ appearing on the RHS. Composing with the folding map ${\mathbb S}^{2}\vee\ldots\vee{\mathbb S}^2\to{\mathbb S}^2$ gives a map $\varphi_L: {\mathbb S}^3\to {\mathbb S}^2$, the homotopy class of which only depends on the homotopy class of the link $L$.
Questions: Provided all this make sense so far, I wonder:
Where is this construction of a map ${\mathbb S}^3\to{\mathbb S}^2$ from a link $L\subset{\mathbb S}^3$ described in the literature?
What's the image of $[\varphi_L]$ under $\pi_3({\mathbb S}^2)\cong{\mathbb Z}$?
Some thoughts on 2: If everything was right so far, the Hopf link should map to the Whitehead product $[\iota_2,\iota_2]$, which is twice the Hopf fibration, so the Hopf link should have value $2$. Moreover, it seems that every $[\varphi_L]$ is stably trivial because the knotting can be undone in one dimension higher, so we should only get even values. Finally, there is a "saddle" homotopy if two strands run side by side in opposite directions, which should reduce the complexity of this significantly.
Motivation: In ordinary type theory, every value of an inductive type is an iterated application of constructors. For higher inductive types in Homotopy Type Theory, this is no longer so, because of the existence of (higher) operations on equality types, the simplest one being equality transitivity $x=y\times y=z\to x=z$. My goal is to give examples which convey the insight that the existence of such (higher) operations is inherently geometric - something that might come as a surprise if one writes them down as maps between iterated equality types, which might look like something quite dry and/or esoteric to do at first. I.e., I am trying to give intuitive understanding why the existence of maps $$\text{refl}\underset{x\underset{X}{=}x}{=}\text{refl}\longrightarrow \text{refl}\underset{\text{refl}\underset{x\underset{X}{=}x}{=}\text{refl}}{=}\text{refl}$$ has something to do with the Hopf fibration. As a first attempt, I want to give a simple geometric description of the Whitehead product $\pi_2(X,x_0)\times\pi_2(X,x_0)\to\pi_3(X,x_0)$ that's amenable to direct translation into Homotopy Type Theory. I am aware that Whitehead products and the Hopf fibration both have been constructed in HoTT, but while very elegant, the constructions I've seen don't strike me as geometrically intuitive.