Understanding the role played by broken symmetry states such as charge, spin, and orbital orders in the mechanism of emergent properties such as high-temperature superconductivity (HTSC) is a major current topic in materials research. That the order may be within one unit cell, such as nematic, was only recently considered theoretically, but its observation in the iron-pnictide and doped cuprate superconductors places it at the forefront of current research. Here we show that the recently discovered BaTi$_2$Sb$_2$O superconductor and its parent compound BaTi$_2$As$_2$O form a symmetry-breaking nematic ground state that can be naturally explained as an intra-unit-cell charge order with $d$-wave symmetry, pointing to the ubiquity of the phenomenon. These findings, together with the key structural features in these materials being intermediate between the cuprate and iron-pnictide HTSC materials, render the titanium oxypnictides an important new material system to understand the nature of nematic order and its relationship to superconductivity.