  We use Higgs bundles to study the 3d $\mathcal{N} = 1$ vacua obtained from
M-theory compactified on a local $Spin(7)$ space given as a four-manifold $M_4$
of ADE singularities with further generic enhancements in the singularity type
along one-dimensional subspaces. There can be strong quantum corrections to the
massless degrees of freedom in the low energy effective field theory, but
topologically robust quantities such as "parity" anomalies are still
calculable. We show how geometric reflections of the compactification space
descend to 3d reflections and discrete symmetries. The "parity" anomalies of
the effective field theory descend from topological data of the
compactification. The geometric perspective also allows us to track various
perturbative and non-perturbative corrections to the 3d effective field theory.
We also provide some explicit constructions of well-known 3d theories,
including those which arise as edge modes of 4d topological insulators, and 3d
$\mathcal{N} = 1$ analogs of grand unified theories. An additional result of
our analysis is that we are able to track the spectrum of extended objects and
their transformations under higher-form symmetries.
