  We use large-scale exact diagonalization to study the quantum Ising chain in
a transverse field with long-range power-law interactions decaying with
exponent $\alpha$. Analyzing various eigenstate and eigenvalue properties, we
find numerical evidence for ergodic behavior in the thermodynamic limit for
$\alpha>0$, \textit{i.~e.} for the slightest breaking of the permutation
symmetry at $\alpha=0$. Considering {an} excited-state fidelity susceptibility,
{an} energy-resolved average level-spacing ratio and the eigenstate
expectations of observables, we observe that a behavior consistent with
eigenstate thermalization first emerges at high energy densities for finite
system sizes, as soon as $\alpha>0$. We argue that ergodicity moves towards
lower energy densities for increasing system sizes. While we argue the system
to be ergodic for any $\alpha>0$, we also find a peculiar behaviour near
$\alpha=2$ suggesting the proximity to a yet unknown integrable point. We
further study the symmetry-breaking properties of the eigenstates. We argue
that for weak transverse fields the eigenstates break the $\mathbb{Z}_2$
symmetry, and show long-range order, at finite excitation energy densities for
all the values of $\alpha$ we can technically address ($\alpha\leq 1.5$). Our
contribution settles central theoretical questions on long-range quantum Ising
chains and are also of direct relevance for the nonequilibrium dynamics in such
systems such as realized experimentally in systems of trapped ions.
