  We describe a mechanism by which the longitudinal thermal conductivity
$\kappa_{xx}$, measured in an in-plane magnetic field, oscillates as a function
of field angle in layered nodal superconductors. These oscillations occur when
the spin-orbit splitting at the nodes is larger than the nodal scattering rate,
and are complementary to vortex-induced oscillations identified previously. In
sufficiently anisotropic materials, the spin-orbit mechanism may be dominant.
As a particular application, we focus on the cuprate high-temperature
superconductor YBa$_2$Cu$_3$O$_{6+x}$. This material belongs to the class of
Rashba bilayers, in which individual CuO$_2$ layers lack inversion symmetry
although the crystal itself is globally centrosymmetric. We show that
spin-orbit coupling endows $\kappa_{xx}/T$ with a characteristic dependence on
magnetic field angle that should be easily detected experimentally, and argue
that for underdoped samples the spin-orbit contribution is larger than the
vortex contribution. A key advantage of the magneto-thermal conductivity is
that it is a bulk probe of spin-orbit physics, and therefore not sensitive to
inversion breaking at surfaces.
