  We study the non-invertible symmetries of class $\mathcal{S}$ theories
obtained by compactifying the type $\mathfrak{a}_{p-1}$ 6d (2,0) theory on a
genus $g$ Riemann surface with no punctures. After setting up the general
framework, we describe how such symmetries can be classified up to genus 5. Of
central interest to us is the question of whether a non-invertible symmetry is
"intrinsic," i.e. whether it can be related to an invertible symmetry by
discrete gauging. We then describe the higher-dimensional origin of our
results, and explain how the Anomaly and Symmetry TFTs, as well as $N$-ality
defects, of class $\mathcal{S}$ theories can be obtained from compactification
of a 7d Chern-Simons theory. Interestingly, we find that the Symmetry TFT for
theories with intrinsically non-invertible symmetries can only be obtained by
coupling the 7d Chern-Simons theory to topological gravity.
