  We investigate the interaction between single quantum emitters and
non-transversally polarised photons for which the electric field vector
amplitude has a significant component in the direction of propagation. Even
though this situation seems to be at odds with the description of light as a
transverse wave, it regularly occurs when interfacing or manipulating emitters
with non-paraxial, guided, or evanescent light. Here, we couple single rubidium
atoms to single photons which are confined in an optical microresonator by
continuous total internal reflection. We experimentally show that the
non-transversal polarisation decisively alters the physics of strong
light--matter interaction. This effect has far-reaching and highly advantageous
implications, which, e.g., significantly advance the field of cavity quantum
electrodynamics with whispering-gallery-mode resonators. In particular, our
finding allows us to experimentally realise a paradigmatic quantum system,
which is ideally suited for basic studies and technological applications.
