  Symmetries of Seiberg-Witten (SW) geometries capture intricate physical
aspects of the underlying 4d $\mathcal{N} = 2$ field theories. For rank-one
theories, these geometries are rational elliptic surfaces whose automorphism
group is a semi-direct product between the Coulomb branch (CB) symmetries and
the Mordell-Weil group. We study quotients of the SW geometry by subgroups of
its automorphism group, which most naturally become gaugings of discrete 0- and
1-form symmetries. Yet, new interpretations of these surgeries become evident
when considering 5d $\mathcal{N}=1$ superconformal field theories. There,
certain CB symmetries are related to symmetries of the corresponding
$(p,q)$-brane web and, as a result, CB surgeries can give rise to (fractional)
S-folds. Another novel interpretation of these quotients is the folding across
dimensions: circle compactifications of the 5d $E_{2N_f + 1}$ Seiberg theories
lead in the infrared to two copies of locally indistinguishable 4d SU(2) SQCD
theories with $N_f$ fundamental flavours. This extends earlier results on
holonomy saddles, while also reproducing detailed computations of 5d BPS
spectra and predicting new 5d and 6d BPS quivers. Finally, we argue that the
semi-direct product structure of the automorphism group of the SW geometry
includes mixed 't Hooft anomalies between the 0- and 1-form symmetries, and we
also present some new results on non-cyclic CB symmetries.
