Revision 2
This note explains how the freeform stretching and rotate with stretch features in Unity's Shuriken ("Legacy") particle system work. It may be useful for those who want to duplicate the effect on the GPU, via Unity's VFX Graph or otherwise. These formulas were obtained through experimentation and, while they seem algebraically equivalent to what Shuriken does, are not the exact operations that Unity uses.
Vector projection of
All these operations are in particle space. That is, we assume that the particle is centered at the origin