  We study a one-parameter family of $\mathcal{N}=2$ anti-de Sitter vacua with
${\rm U}(1)^2$ symmetry of gauged four-dimensional maximal supergravity, with
dyonic gauge group $[{\rm SO}(6)\times {\rm SO}(1,1)]\ltimes \mathbb{R}^{12}$.
These backgrounds are known to correspond to Type IIB S-fold solutions with
internal manifold of topology $S^1\times S^5$. The family of AdS$_4$ vacua is
parametrized by a modulus $\chi$. Although $\chi$ appears non-compact in the
four-dimensional supergravity, we show that this is just an artefact of the
four-dimensional description. We give the 10-dimensional geometric
interpretation of the modulus and show that it actually has periodicity of
$\frac{2\pi}{T}$, which is the inverse radius of $S^1$. We deduce this by
providing the explicit $D=10$ uplift of the family of vacua as well as
computing the entire modulus-dependent Kaluza-Klein spectrum as a function of
$\chi$. At the special values $\chi=0$ and $\chi=\frac{\pi}{T}$, the symmetry
enhances according to ${\rm U}(1)^2\rightarrow{\rm U}(2)$, giving rise however
to inequivalent Kaluza-Klein spectra. At $\chi=\frac{\pi}{T}$, this realizes a
bosonic version of the "space invaders" scenario with additional massless
vector fields arising from formerly massive fields at higher Kaluza-Klein
levels.
