Context. Molecular hydrogen (H₂) is the most abundant molecule in the interstellar medium. Because of its excited form in irradiated regions, it is a useful tool for studying photodissociation regions (PDRs), where radiative feedback from massive stars on molecular clouds is dominant. The James Webb Space Telescope (JWST), with its high spatial resolution, sensitivity, and wavelength coverage, provides unique access to the detection of most of the H₂ rotational and rovibrational lines, as well as the analysis of their spatial morphology. Aims. Our goal is to use H₂ line emission detected with JWST in the Horsehead nebula to constrain the physical parameters (e.g., extinction, gas temperature, and thermal pressure) throughout the PDR and its geometry. Methods. We used spectro-imaging data acquired using both the NIRSpec and MIRI-MRS instruments on board JWST to study the H₂ spatial distribution at very small scales (down to 0.1^′′). From the H₂ line ratios, we constrained the extinction throughout the PDR. We then studied the excitation of H₂ levels in detail and used this analysis to derive the physical parameters. Results. We detect hundreds of H₂ rotational and rovibrational lines in the Horsehead nebula. The H₂ morphology reveals a spatial separation between H₂ lines (∼0.5^′′) across the PDR interface. Far-ultraviolet (FUV)-pumped lines (v = 0 J _u > 6, v > 0) peak closer to the edge of the PDR than thermalized lines. From H₂ lines arising from the same upper level, we estimated the value of extinction throughout the PDR. We find that A _V increases from the edge of the PDR to the second and third H₂ filaments. We find A _V =0.3 ± 1.3 in the first filament and A _V =6.1 ± 1.4 in the second and third filaments. We then studied the H₂ excitation in different regions across the PDR. The excitation diagrams were fit by two excitation temperatures. As the first levels of H₂ are thermalized, the colder temperature corresponds to the gas temperature. The second, hotter component corresponds to the FUV-pumped levels. In each filament, we derive a gas temperature of T ∼500 K. The temperature profile shows that the observed gas temperature remains nearly constant throughout the PDR, with a slight decrease in each of the dissociation fronts. The spatial distribution of H₂ reveals that most of the H₂ column density is concentrated in the second and third filaments. The column density in the first filament is approximately N(H₂)=(3.8 ± 0.8) × 10¹⁹ cm⁻², while in the second and third filaments it is N(H₂)=(1.9 ± 0.4) × 10²⁰ cm⁻², about five times higher. The ortho-to-para ratio (OPR) is far from equilibrium, varying from 2–2.5 at the edge of each dissociation front to 1.3–1.5 deeper into the PDR. We observe a clear spatial separation between the para and ortho rovibrational levels, as well as between 0−0 S(2) and 0−0 S(1), indicating that efficient ortho-para conversion and preferential ortho self-shielding are driving the spatial variations of the OPR. Finally, we derive a thermal pressure in the first filament of about P _gas ≥ 6 × 10⁶ K cm⁻³, which is approximately ten times higher than that of the ionized gas. We highlight that template stationary 1D PDR models cannot account for the intrinsic 2D structure and the very high temperature observed in the Horsehead nebula. We argue that the highly excited, over-pressurized H₂ gas at the edge of the PDR interface could originate from mixing between the cold and hot phases induced by photo-evaporation of the cloud. Conclusions. The analysis of H₂ lines detected with JWST provides unique access to the geometry and physical conditions in the Horsehead nebula at very small scales and reveals, for the first time, the possible importance of dynamical effects at the edge of the PDR. This study nevertheless highlights the need for extended modeling of these dynamical effects.