It is a variational method, and consists of special homotopies (they are shortest paths with respect to Willmore energy); contrast with Thurston's corrugations, which are generic.
Rob Kusner proposed optimal eversions using the Willmore energy on the space of all immersions of the sphere
The round sphere and the inside-out round sphere are the unique global minima for Willmore energy, and a minimax eversion is a path connecting these by passing over a saddle point (like traveling between two valleys via a mountain pass).
[2] Kusner's half-way models are saddle points for Willmore energy, arising (according to a theorem of Bryant) from certain complete minimal surfaces in 3-space; the minimax eversions consist of gradient ascent from the round sphere to the half-way model, then gradient descent down (gradient descent for Willmore energy is called Willmore flow).
More symmetrically, start at the half-way model; push in one direction and follow Willmore flow down to a round sphere; push in the opposite direction and follow Willmore flow down to the inside-out round sphere.