  We show with first-principles molecular dynamics the persistence of intrinsic
$\langle111\rangle$ Ti off-centerings for BaTiO$_3$ in its cubic paraelectric
phase. Intriguingly, these are inconsistent with the Pm$\bar 3$m space group
often used to atomistically model this phase using density functional theory or
similar methods. Therefore we deploy a systematic symmetry analysis to
construct representative structural models in the form of supercells that
satisfy a desired point symmetry but are built from the combination of
lower-symmetry primitive cells. We define as structural prototypes the smallest
of these that are both energetically and dynamically stable. Remarkably, two
40-atom prototypes can be identified for paraelectric BaTiO$_3$; these are also
common to many other $AB$O$_3$ perovskites. These prototypes can offer
structural models of paraelectric phases that can be used for the computational
engineering of functional materials. Last, we show that the emergence of
B-cation off-centerings and the primitive-cell phonon instabilities are
controlled by the equilibrium volume, in turn dictated by the filler A cation.
