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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 H0 while the scattering twin baryons can reduce the matter fluctuations S8, which helps to relax the observed H0 and S8 tensions simultaneously. This scenario is different from the typical ΛCDM + ∆Neff model, in which extra radiation helps with the Hubble tension but worsens the S8 tension. For instance, when including the SH0ES and 2013 Planck SZ data in the fit, we find that a universe with ≳ 20% of the dark matter comprised of twin baryons is preferred over ΛCDM by ∼ 4σ. If the twin sector is indeed responsible for resolving the H0 and S8 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.