  Monolayers of transition metal dichalcogenides (TMDs) in the 2H structural
phase have been recently classified as higher-order topological insulators
(HOTI), protected by $C_3$ rotation symmetry. In addition, theoretical
calculations show an orbital Hall plateau in the insulating gap of TMDs,
characterized by an orbital Chern number. We explore the correlation between
these two phenomena in TMD monolayers in two structural phases: the
noncentrosymmetric 2H and the centrosymmetric 1T. Using density functional
theory, we confirm the characteristics of 2H-TMDs and reveal that 1T-TMDs are
identified by a $\mathbb{Z}_4$ topological invariant. As a result, when cut
along appropriate directions, they host conducting edge-states, which cross
their bulk energy-band gaps and can transport orbital angular momentum. Our
linear response calculations thus indicate that the HOTI phase is accompanied
by an orbital Hall effect. Using general symmetry arguments, we establish a
connection between the two phenomena with potential implications for
spin-orbitronics.
