  Type IIB S-folds of the form $\textrm{AdS}_{4} \times \textrm{S}^1 \times
\textrm{S}^5$ have been shown to contain axion-like deformations parameterising
flat directions in the 4D scalar potential and corresponding to marginal
deformations of the dual S-fold CFT's. In this note we present a
group-theoretical characterisation of such flat deformations and provide a 5D
interpretation thereof in terms of $\mathfrak{so}(6)$-valued duality twists
inducing a class of Cremmer--Scherk--Schwarz flat gaugings in a reduction from
5D to 4D. In this manner we establish the existence of two flat deformations
for the $\mathcal{N}=4$ and $\textrm{SO}(4)$ symmetric S-fold causing a
symmetry breaking down to its $\textrm{U}(1)^2$ Cartan subgroup. The result is
a new two-parameter family of non-supersymmetric S-folds which are
perturbatively stable at the lower-dimensional supergravity level, thus
providing the first examples of such type IIB backgrounds.
