The Paleocene-Eocene Thermal Maximum (PETM; 56 Ma) is one of our best geological analogs for understanding climate dynamics in a "greenhouse" world. However, proxy data representing the event are only available from select marine and terrestrial sedimentary sequences that are unevenly distributed across Earth's surface, limiting our view of the spatial patterns of climate change. Here, we use paleoclimate data assimilation (DA) to combine climate model and proxy information and create a spatially complete reconstruction of the PETM and the climate state that precedes it ("PETM-DA"). Our data-constrained results support strong polar amplification, which in the absence of an extensive cryosphere, is related to temperature feedbacks and loss of seasonal snow on land. The response of the hydrological cycle to PETM warming consists of a narrowing of the Intertropical Convergence Zone, off-equatorial drying, and an intensification of seasonal monsoons and winter storm tracks. Many of these features are also seen in simulations of future climate change under increasing anthropogenic emissions. Since the PETM-DA yields a spatially complete estimate of surface air temperature, it yields a rigorous estimate of global mean temperature change (5.6 ∘C; 5.4 ∘C to 5.9 ∘C, 95% CI) that can be used to calculate equilibrium climate sensitivity (ECS). We find that PETM ECS was 6.5 ∘C (5.7 ∘C to 7.4 ∘C, 95% CI), which is much higher than the present-day range. This supports the view that climate sensitivity increases substantially when greenhouse gas concentrations are high.