  Spin-driven nematicity, or the breaking of the point-group symmetry of the
lattice without long-range magnetic order, is clearly quite important in
iron-based superconductors. From a symmetry point of view, nematic order can be
described as a coherent locking of spin fluctuations in two interpenetrating
N\'eel sublattices with ensuing nearest-neighbor bond order and an absence of
static magnetism. Here, we argue that the low-temperature state of the recently
discovered superconductor BaTi$_2$Sb$_2$O is a strong candidate for a more
exotic form of spin-driven nematic order, in which fluctuations occurring in
four N\'eel sublattices promote both nearest- and next-nearest neighbor bond
order. We develop a low-energy field theory of this state and show that it can
have, as a function of temperature, up to two separate bond-order phase
transitions -- namely, one that breaks rotation symmetry and one that breaks
reflection and translation symmetries of the lattice. The resulting state has
an orthorhombic lattice distortion, an intra-unit-cell charge density wave, and
no long-range magnetic order, all consistent with reported measurements of the
low-temperature phase of BaTi$_2$Sb$_2$O. We then use density functional theory
calculations to extract exchange parameters to confirm that the model is
applicable to BaTi$_2$Sb$_2$O.
