  Inversion symmetry breaking is a ubiquitous concept in condensed-matter
science. On the one hand, it is a prerequisite for many technologically
relevant effects such as piezoelectricity, photovoltaic and nonlinear optical
properties and spin-transport phenomena. On the other hand, it may determine
abstract properties such as the electronic topology in quantum materials.
Therefore, the creation of materials where inversion symmetry can be turned on
or off by design may be the ultimate route towards controlling parity-related
phenomena and functionalities. Here, we engineer the symmetry of ultrathin
epitaxial oxide films by sub-unit-cell growth control. We reversibly activate
and deactivate inversion symmetry in the layered hexagonal manganites,
h-RMnO$_3$ with R = Y, Er, Tb. While an odd number of half-unit-cell layers
exhibits a breaking of inversion symmetry through its arrangement of MnO$_5$
bipyramids, an even number of such half-unit-cell layers takes on a
centrosymmetric structure. Here we control the resulting symmetry by tracking
the growth in situ via optical second-harmonic generation. We furthermore
demonstrate that our symmetry engineering works independent of the choice of R
and even in heterostructures mixing constituents with different R in a
two-dimensional growth mode. Symmetry engineering on the atomic level thus
suggests a new platform for the controlled activation and deactivation of
symmetry-governed functionalities in oxide-electronic epitaxial thin films.
