  Quasiperiodic modulation can prevent isolated quantum systems from
equilibrating by localizing their degrees of freedom. In this article, we show
that such systems can exhibit dynamically stable long-range orders forbidden in
equilibrium. Specifically, we show that the interplay of symmetry breaking and
localization in the quasiperiodic quantum Ising chain produces a
\emph{quasiperiodic Ising glass} stable at all energy densities. The glass
order parameter vanishes with an essential singularity at the melting
transition with no signatures in the equilibrium properties. The zero
temperature phase diagram is also surprisingly rich, consisting of
paramagnetic, ferromagnetic and quasiperiodically alternating ground state
phases with extended, localized and critically delocalized low energy
excitations. The system exhibits an unusual quantum Ising transition whose
properties are intermediate between those of the clean and infinite randomness
Ising transitions. Many of these results follow from a geometric generalization
of the Aubry-Andr\'e duality which we develop. The quasiperiodic Ising glass
may be realized in near term quantum optical experiments.
