  Pseudorelativistic Dirac quasiparticles have emerged in a plethora of
artificial graphene systems that mimic the underlying honeycomb symmetry of
graphene. However, it is notoriously difficult to manipulate their properties
without modifying the lattice structure. Here we theoretically investigate
polaritons supported by honeycomb metasurfaces and, despite the trivial nature
of the resonant elements, we unveil rich Dirac physics stemming from a
non-trivial winding in the light-matter interaction. The metasurfaces
simultaneously exhibit two distinct species of massless Dirac polaritons,
namely type-I and type-II. By modifying only the photonic environment via an
enclosing cavity, one can manipulate the location of the type-II Dirac points,
leading to qualitatively different polariton phases. This enables one to alter
the fundamental properties of the emergent Dirac polaritons while preserving
the lattice structure - a unique scenario which has no analog in real or
artificial graphene systems. Exploiting the photonic environment will thus give
rise to unexplored Dirac physics at the subwavelength scale.
