  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 K\"ahler 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.
