  We study extensions of the Twin Higgs model that solve the Hierarchy problem
and simultaneously address problems of the large- and small-scale structures of
the Universe. Besides naturally providing dark matter (DM) candidates as the
lightest charged twin fermions, the twin sector contains a light photon and
neutrinos, which can modify structure formation relative to the prediction from
the $\Lambda$CDM paradigm. We focus on two scenarios. First, we study a
Fraternal Twin Higgs model in which the spin-3/2 baryon
$\hat{\Omega}\sim(\hat{b}\hat{b}\hat{b})$ and the lepton twin tau $\hat{\tau}$
contribute to the dominant and subcomponent dark matter densities. A
non-decoupled scattering between the twin tau and twin neutrino arising from a
gauged twin lepton number symmetry provides a drag force that damps the density
inhomogeneity of a dark matter subcomponent. Next, we consider the possibility
of having the twin hydrogen atom $\hat{H}$ as the dominant DM component. After
recombination, a small fraction of the twin protons and leptons remains ionized
during structure formation, and their scattering to twin neutrinos through a
gauged U$(1)_{B-L}$ force provides the mechanism that damps the density
inhomogeneity. Both scenarios realize the Partially Acoustic dark matter
(PAcDM) scenario and explain the $\sigma_8$ discrepancy between the CMB and
weak lensing results. Moreover, the self-scattering neutrino behaves as a dark
fluid that enhances the size of the Hubble rate $H_0$ to accommodate the local
measurement result while satisfying the CMB constraint. For the small-scale
structure, the scattering of $\hat{\Omega}$'s and $\hat{H}$'s through the twin
photon exchange generates a self-interacting dark matter (SIDM) model that
solves the mass deficit problem from dwarf galaxy to galaxy cluster scales.
