  We study maximal representations of surface groups
$\rho:\pi_1(\Sigma)\to\mathrm{SO}_0(2,n+1)$ via the introduction of
$\rho$-invariant pleated surfaces inside the pseudo-Riemannian space
$\mathbb{H}^{2,n}$ associated to maximal geodesic laminations of $\Sigma$.
  We prove that $\rho$-invariant pleated surfaces are always embedded, acausal,
and possess an intrinsic pseudo-metric and a hyperbolic structure. We describe
the latter by constructing a shear cocycle from the cross ratio naturally
associated to $\rho$. The process developed to this purpose applies to a wide
class of cross ratios, including examples arising from Hitchin and
$\Theta$-positive representations in $\mathrm{SO}(p,q)$. We also show that the
length spectrum of $\rho$ dominates the ones of $\rho$-invariant pleated
surfaces, with strict inequality exactly on curves that intersect the bending
locus.
  We observe that the canonical decomposition of a $\rho$-invariant pleated
surface into leaves and plaques corresponds to a decomposition of the
Guichard-Wienhard domain of discontinuity of $\rho$ into standard fibered
blocks, namely triangles and lines of photons. Conversely, we give a concrete
construction of photon manifolds fibering over hyperbolic surfaces by gluing
together triangles of photons.
  The tools we develop allow to recover various results by Collier, Tholozan,
and Toulisse on the (pseudo-Riemannian) geometry of $\rho$ and on the
correspondence between maximal representations and fibered photon manifolds
through a constructive and geometric approach, bypassing the use of Higgs
bundles.
