Start with the Dynkin diagram of an irreducible root system, typically associated with a simple Lie algebra over $\mathbb{C}$ or a simple algebraic group. Most of the simply-laced ADE diagrams admit natural symmetries of order 2: type $A_r$ when $r \geq 2$, type $D_r$ when $r \geq 4$, and type $E_6$. In the case of type $D_4$, there is also a symmetry of order 3. There is an obvious way to "fold" each diagram by identifying those vertices which are interchanged by the symmetry. This yields Dynkin diagrams with two root lengths. For example, $E_6 \leadsto F_4$.

What I don't see is any general theory dictating in advance how to interpret this process.

In particular, does a pair of simple roots (vertices) interchanged by the symmetry in type $E_6$
lead to a long or a short simple root in type $F_4$? In effect, there are (Langlands) dual
possibilities, switching long and short roots while also switching types $B_r$ and $C_r$. It may be difficult to formulate a single theory of folding and duality which makes all instances
look natural, but it's at least reasonable to ask:

What is the origin of the notion of "folding" for Dynkin diagrams?

It's well-known that the ADE graphs are ubiquitous in various areas of mathematics; for an
older survey see the paper by Hazewinkel et al.
here. The usual
convention is to regard all vertices of the graph as corresponding to *long* simple roots.
In their examples, the most common type of folding occurs this way:
$A_{2r-1} \leadsto C_r, \: D_{r+1} \leadsto B_r, \: D_4 \leadsto G_2, \: E_6 \leadsto F_4$,
where multiple simple roots are always taken to *short* simple roots. Examples occur in
various parts of Lie theory: classification of real forms of complex semisimple Lie algebras,
classification of $k$-forms of reductive $k$-isotropic algebraic groups (Satake, Tits), etc.

But in the study of simple singularities of types ADE (work of Grothendieck, Brieskorn, Slodowy),
adapted to other Lie types, the natural folding procedure is dual to the above one:
$A_{2r-1} \leadsto B_r, \: D_{r+1} \leadsto C_r, \: D_4 \leadsto G_2, \: E_6 \leadsto F_4$,
where now multiple simple roots lead to *long* simple roots.

My own motivation comes from related work on representations of Lie algebras of simple algebraic groups in prime characteristic. Here the most elusive non-restricted representations belong to nilpotent elements (or orbits). The case of regular nilpotents is easy, but already the subregular case is too difficult for current algebraic methods to complete in case of two root lengths. (For large enough primes, the deep work of Bezrukavnikov, Mirkovic, and Rumynin has pushed farther.)

It's instructive to think of (dual) folding in connection with Dynkin curves (Springer fibers
in the subregular case) and the detailed results of
J.C. Jantzen in *Subregular nilpotent representations of Lie algebras in prime characteristic*,
Represent. Theory 3 (1999), 153-222, freely available
here. His incomplete
dimension calculations involving two root lengths would become transparent if the folding process could be rigorously justified in advance.