  The mirror twin Higgs model (MTH) is a solution to the Higgs hierarchy
problem that provides well-predicted cosmological signatures with only three
extra parameters: the temperature of the twin sector, the abundance of twin
baryons, and the vacuum expectation value (VEV) of twin electroweak symmetry
breaking. These parameters specify the behavior of twin radiation and the
acoustic oscillations of twin baryons, which lead to testable effects on the
cosmic microwave background (CMB) and large-scale structure (LSS). While
collider searches can only probe the twin VEV, through a fit to cosmological
data we show that the existing CMB (Planck18 TTTEEE+lowE+lowT+lensing) and LSS
(KV450) data already provide useful constraints on the remaining MTH
parameters. Additionally, we show that the presence of twin radiation in this
model can raise the Hubble constant $H_0$ while the scattering twin baryons can
reduce the matter fluctuations $S_8$, which helps to relax the observed $H_0$
and $S_8$ tensions simultaneously. This scenario is different from the typical
$\Lambda$CDM + $\Delta N_{\rm eff}$ model, in which extra radiation helps with
the Hubble tension but worsens the $S_8$ tension. For instance, when including
the SH0ES and Planck SZ data in the fit, we find that a universe with $\gtrsim
20\%$ of the dark matter comprised of twin baryons is preferred over
$\Lambda$CDM by $\sim4\sigma$. If the twin sector is indeed responsible for
resolving the $H_0$ and $S_8$ tensions, future measurements from the Euclid
satellite and CMB Stage 4 experiment will further measure the twin parameters
to $O(1-10\%)$-level precision. Our study demonstrates how models with hidden
naturalness can potentially be probed using precision cosmological data.
