  In recent years, transition metal dichalcogenides (TMDs) have garnered great
interest as topological materials -- monolayers of centrosymmetric
$\beta$-phase TMDs have been identified as 2D topological insulators (TIs), and
bulk crystals of noncentrosymmetric $\gamma$-phase MoTe$_2$ and WTe$_2$ have
been identified as type-II Weyl semimetals. However, ARPES and STM probes of
these TMDs have revealed huge, "arc-like" 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 use first-principles calculations
and (nested) Wilson loops to analyze the bulk and surface electronic structure
of both $\beta$- and $\gamma$-MoTe$_2$, finding that $\beta$-MoTe$_2$
($\gamma$-MoTe$_2$ gapped with symmetry-preserving distortion) is an
inversion-symmetry-indicated $\mathbb{Z}_{4}$-nontrivial ($noncentrosymmetric,
non$-$symmetry$-$indicated$) higher-order TI (HOTI) driven by double band
inversion. Both structural phases of MoTe$_2$ exhibit the same surface features
as WTe$_2$, revealing that the large Fermi arcs are in fact not topologically
trivial, but are rather the characteristic split and gapped fourfold surface
states of a HOTI. We also show that, when the effects of SOC are neglected,
$\beta$-MoTe$_2$ is a nodal-line semimetal with $\mathbb{Z}_{2}$-nontrivial
monopole nodal lines (MNLSM). This finding confirms that MNLSMs driven by
double band inversion are the weak-SOC 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.
