  In this paper we study the para-hyperK\"ahler geometry of the deformation
space of MGHC anti-de Sitter structures on $\Sigma\times\mathbb R$, for
$\Sigma$ a closed oriented surface. We show that a neutral pseudo-Riemannian
metric and three symplectic structures coexist with an integrable complex
structure and two para-complex structures, satisfying the relations of
para-quaternionic numbers. We show that these structures are directly related
to the geometry of MGHC manifolds, via the Mess homeomorphism, the
parameterization of Krasnov-Schlenker by the induced metric on $K$-surfaces,
the identification with the cotangent bundle $T^*\mathcal{T}(\Sigma)$, and the
circle action that arises from this identification. Finally, we study the
relation to the natural para-complex geometry that the space inherits from
being a component of the $\mathrm{PSL}(2,\mathbb{B})$-character variety, where
$\mathbb{B}$ is the algebra of para-complex numbers, and the symplectic
geometry deriving from Goldman symplectic form.
