  We present the $\nu\phi$MTH, a Mirror Twin Higgs (MTH) model realizing
asymmetric reheating, baryogenesis and twin-baryogenesis through the
out-of-equilibrium decay of a right-handed neutrino without any hard
$\mathbb{Z}_2$ breaking. The MTH is the simplest Neutral Naturalness solution
to the little hierarchy problem and predicts the existence of a twin dark
sector related to the Standard Model (SM) by a $\mathbb{Z}_2$ symmetry that is
only softly broken by a higher twin Higgs vacuum expectation value. The
asymmetric reheating cools the twin sector compared to the visible one, thus
evading cosmological bounds on $\Delta N_{\mathrm{eff}}$. The addition of
(twin-)colored scalars allows for the generation of the visible baryon
asymmetry and, by the virtue of the $\mathbb{Z}_2$ symmetry, also results in
the generation of a twin baryon asymmetry. We identify a unique scenario with
top-philic couplings for the new scalars that can satisfy all cosmological,
proton decay and LHC constraints; yield the observed SM baryon asymmetry; and
generate a wide range of possible twin baryon DM fractions, from negligible to
unity. The viable regime of the theory contains several hints as to the
possible structure of the Twin Higgs UV completion. Our results motivate the
search for the rich cosmological and astrophysical signatures of twin baryons,
and atomic dark matter more generally, at cosmological, galactic and stellar
scales.
