Orbital symmetry of charge density wave order in La$_{1.875}$Ba$_{0.125}$CuO$_4$ and YBa$_2$Cu$_3$O$_{6.67}$

Recent theories of charge density wave (CDW) order in high temperature superconductors have predicted a primarily $d$ CDW orbital symmetry. Here, we report on the orbital symmetry of CDW order in the canonical cuprate superconductors La$_{1.875}$Ba$_{0.125}$CuO$_4$ (LBCO) and YBa$_2$Cu$_3$O$_{6.67}$ (YBCO), using resonant soft x-ray scattering and a model mapped to the CDW orbital symmetry. From measurements sensitive to the O sublattice, we conclude that LBCO has predominantly $s'$ CDW orbital symmetry, in contrast to the $d$ orbital symmetry recently reported in other cuprates. Additionally, we find that the C$_4$ orbital symmetry of the Cu sublattice scattering is approximately preserved in LBCO and broken in YBCO. This work highlights orbital symmetry as an additional key property of CDW order that distinguishes the different cuprate families. We discuss how the CDW symmetry may be related to the ``1/8--anomaly'' and to static spin ordering.