  We investigate the Autler-Townes splitting for Rydberg atoms dressed with
linearly polarized microwave radiation, resonant with generic
$S_{1/2}\leftrightarrow{P}_{1/2}$ and $S_{1/2}\leftrightarrow{P}_{3/2}$
transitions. The splitting is probed using laser light via
electromagnetically-induced transparency measurements, where the transmission
of probe laser light reveals a two-peak pattern. In particular, this pattern is
invariant under rotation of the microwave field polarization. In consequence,
we establish $S \leftrightarrow P$ Rydberg transitions as ideally suited for
polarization-insensitive electrometry, contrary to recent findings [A.
Chopinaud and J.D. Pritchard, Phys. Rev. Appl. $\mathbf{16}$, 024008 (2021)].
