  Primordial magnetic fields are often thought to be the early Universe seeds
that have bloomed into what we observe today as galactic and extra-galactic
magnetic fields. Owing to their minuscule strength, primordial magnetic fields
are very hard to detect in cosmological and astrophysical observations. We show
how this changes if a part of neutral Dark Matter has a magnetic
susceptibility. In this way, by studying Dark Matter one can obtain information
about the properties of primordial magnetic fields, even if the latter have a
comoving amplitude $B_0 \lesssim0.01~\mbox{nG}$. In our model Dark Matter is a
stable singlet scalar $\chi$, which interacts with electromagnetism through the
Rayleigh operator as $\chi^2 F_{\mu \nu} F^{\mu \nu}/\Lambda^2$. For primordial
magnetic fields present in the early Universe this operator forces the
$Z_2$-symmetry of the model to be spontaneously broken. Later, when the
primordial magnetic field redshifts below a critical value, the symmetry is
restored through an "inverse phase transition". At that point the field $\chi$
begins to oscillate and acts as a "magnetomorphic" Dark Matter component,
inheriting the properties of the primordial magnetic field space distribution.
In particular, for a nearly flat spectrum of magnetic field fluctuations, the
scalar $\chi$ carries a statistically anisotropic isocurvature mode. We discuss
the parameter space of the model and consider the possibility that the bulk of
the Dark Matter is composed of the same particles $\chi$ produced via the
freeze-in mechanism.
