  We study the Klein four-group $(K_4)$ symmetry of the time-dependent
Schr\"odinger equation for the conformal mechanics model of de
Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling
constant $g=\nu(\nu+1)\geq -1/4$. We show that it undergoes a complete or
partial (at half-integer $\nu$) breaking on eigenstates of the system, and is
the automorphism of the $\mathfrak{osp}(2,2)$ superconformal symmetry in
super-extensions of the model by inducing a transformation between the exact
and spontaneously broken phases of $\mathcal{N}=2$ Poincar\'e supersymmetry. We
exploit the $K_4$ symmetry and its relation with the conformal symmetry to
construct the dual Darboux transformations which generate spectrally shifted
pairs of the rationally deformed AFF models. Two distinct pairs of intertwining
operators originated from Darboux duality allow us to construct complete sets
of the spectrum generating ladder operators that identify specific finite-gap
structure of a deformed system and generate three distinct related versions of
nonlinearly deformed $\mathfrak{sl}(2,{\mathbb R})$ algebra as its symmetry. We
show that at half-integer $\nu$, the Jordan states associated with confluent
Darboux transformations enter the construction, and the spectrum of rationally
deformed AFF systems undergoes structural changes.
