We consider Little String Theories (LSTs) that are engineered by $N$ parallel M5-branes probing a transverse $\mathbb{Z}_M$ geometry. By exploiting a dual description in terms of F-theory compactified on a toric Calabi-Yau threefold $X_{N,M}$, we establish numerous symmetries that leave the BPS partition function $\mathcal{Z}_{N,M}$ invariant. They furthemore act in a non-perturbative fashion from the point of view of the low energy quiver gauge theory associated with the LST. We present different group theoretical organisations of these symmetries, thereby generalising the results of [arXiv:1811.03387] to the case of generic $M \geq 1$. We also provide a Mathematica package that allows to represent them in terms of matrices that act linearly on the Kähler parameters of $X_{N,M}$. From the perspective of dual realisations of the LSTs the symmetries found here act in highly nontrivial ways: as an example, we consider a formulation of $\mathcal{Z}_{N,M}$ in terms of correlation functions of a vertex operator algebra, whose commutation relations are governed by an affine quiver algebra. We show the impact of the symmetry transformations on the latter and discuss invariance of $\mathcal{Z}_{N,M}$ from this perspective for concrete examples.