  We say that $\Gamma$, the boundary of a bounded Lipschitz domain, is locally
dilation invariant if, at each $x\in \Gamma$, $\Gamma$ is either locally $C^1$
or locally coincides (in some coordinate system centred at $x$) with a
Lipschitz graph $\Gamma_x$ such that $\Gamma_x=\alpha_x\Gamma_x$, for some
$\alpha_x\in (0,1)$. In this paper we study, for such $\Gamma$, the essential
spectrum of $D_\Gamma$, the double-layer (or Neumann-Poincar\'e) operator of
potential theory, on $L^2(\Gamma)$. We show, via localisation and
Floquet-Bloch-type arguments, that this essential spectrum is the union of the
spectra of related continuous families of operators $K_t$, for $t\in
[-\pi,\pi]$; moreover, each $K_t$ is compact if $\Gamma$ is $C^1$ except at
finitely many points. For the 2D case where, additionally, $\Gamma$ is
piecewise analytic, we construct convergent sequences of approximations to the
essential spectrum of $D_\Gamma$; each approximation is the union of the
eigenvalues of finitely many finite matrices arising from Nystr\"om-method
approximations to the operators $K_t$. Through error estimates with explicit
constants, we also construct functionals that determine whether any particular
locally-dilation-invariant piecewise-analytic $\Gamma$ satisfies the well-known
spectral radius conjecture, that the essential spectral radius of $D_\Gamma$ on
$L^2(\Gamma)$ is $<1/2$ for all Lipschitz $\Gamma$. We illustrate this theory
with examples; for each we show that the essential spectral radius is $<1/2$,
providing additional support for the conjecture. We also, via new results on
the invariance of the essential spectral radius under locally-conformal
$C^{1,\beta}$ diffeomorphisms, show that the spectral radius conjecture holds
for all Lipschitz curvilinear polyhedra.
