  Each unit cell in YBa$_2$Cu$_3$O$_{6+x}$ contains a pair of two-dimensional
CuO$_2$ layers. While the crystal structure is globally inversion symmetric,
the individual layers are not. This leads, necessarily, to a nonvanishing
Rashba spin-orbit coupling (SOC) in the CuO$_2$ layers, with opposite signs of
the coupling constant in each layer. These so-called Rashba bilayers generate
hidden spin textures, with a vansishing net spin at each $k$-point in the
Brillouin zone, but nonvanishing spin textures in each layer separately. Here,
we trace the microscopic origin of the Rashba splitting through the orbital
structure of the CuO$_2$ conduction bands, obtain a generic three-orbital model
Hamiltonian, and show that the magnitude of the spin-splitting predicted by
density functional theory is $\sim 10$~meV.
