  We build on previous studies of the Higgs and Coulomb branches of SUSY quiver
theories having 8 supercharges, including $3d~{\cal N}=4$, and Classical gauge
groups. The vacuum moduli spaces of many such theories can be parameterised by
pairs of nilpotent orbits of Classical Lie algebras; they are transverse to one
orbit and intersect the closure of the second. We refer to these transverse
spaces as Slodowy intersections. They embrace reduced single instanton moduli
spaces, nilpotent orbits, Kraft-Procesi transitions and Slodowy slices, as well
as other types. We show how quiver subtractions, between multi-flavoured
unitary or ortho-symplectic quivers, can be used to find a complete set of
Higgs branch constructions for the Slodowy intersections of any Classical
group. We discuss the relationships between the Higgs and Coulomb branches of
these quivers and $T_{\sigma}^{\rho}$ theories in the context of $3d$ mirror
symmetry, including problematic aspects of Coulomb branch constructions from
ortho-symplectic quivers. We review Coulomb and Higgs branch constructions for
a subset of Slodowy intersections from multi-flavoured Dynkin diagram quivers.
We tabulate Hilbert series and Highest Weight Generating functions for Slodowy
intersections of Classical algebras up to rank 4. The results are confirmed by
direct calculation of Hilbert series from a localisation formula for normal
Slodowy intersections that is related to the Hall Littlewood polynomials.
