In recent years, transition metal dichalcogenides (TMDs) have garnered great interest as topological materials. In particular, monolayers of centrosymmetric β-phase TMDs have been identified as 2D topological insulators (TIs), and bulk crystals of noncentrosymmetric γ-phase MoTe_{2} and WTe_{2} have been identified as type-II Weyl semimetals. However, angle-resolved photoemission spectroscopy and STM probes of these semimetals have revealed huge, arclike surface states that overwhelm, and are sometimes mistaken for, the much smaller topological surface Fermi arcs of bulk type-II Weyl points. In this Letter, we calculate the bulk and surface electronic structure of both β- and γ-MoTe_{2}. We find that β-MoTe_{2} is, in fact, a Z_{4}-nontrivial higher-order TI (HOTI) driven by double band inversion and exhibits the same surface features as γ-MoTe_{2} and γ-WTe_{2}. We discover that these surface states are not topologically trivial, as previously characterized by the research that differentiated them from the Weyl Fermi arcs but, rather, are the characteristic split and gapped fourfold Dirac surface states of a HOTI. In β-MoTe_{2}, this indicates that it would exhibit helical pairs of hinge states if it were bulk insulating, and in γ-MoTe_{2} and γ-WTe_{2}, these surface states represent vestiges of HOTI phases without inversion symmetry that are nearby in parameter space. Using nested Wilson loops and first-principles calculations, we explicitly demonstrate that, when the Weyl points in γ-MoTe_{2} are annihilated, which may be accomplished by symmetry-preserving strain or lattice distortion, γ-MoTe_{2} becomes a nonsymmetry-indicated, noncentrosymmetric HOTI. We also show that, when the effects of spin-orbit coupling are neglected, β-MoTe_{2} is a nodal-line semimetal with Z_{2}-nontrivial monopole nodal lines (MNLSM). This finding confirms that MNLSMs driven by double band inversion are the weak-spin-orbit coupling limit of HOTIs, implying that MNLSMs are higher-order topological semimetals with flat-band-like hinge states, which we find to originate from the corner modes of 2D "fragile" TIs.