$Context$. Pulsar wind nebulae (PWNe) represent the most prominent population of Galactic very-high-energy gamma-ray sources and are thought to be an efficient source of leptonic cosmic rays. Vela X is a nearby middle-aged PWN, which shows bright X-ray and TeV gamma-ray emission towards an elongated structure called the cocoon. $Aims$. Since TeV emission is likely inverse-Compton emission of electrons, predominantly from interactions with the cosmic microwave background, while X-ray emission is synchrotron radiation of the same electrons, we aim to derive the properties of the relativistic particles and of magnetic fields with minimal modelling. $Methods$. We used data from the $Suzaku$ XIS to derive the spectra from three compact regions in Vela X covering distances from 0.3 to 4 pc from the pulsar along the cocoon. We obtained gamma-ray spectra of the same regions from H.E.S.S. observations and fitted a radiative model to the multi-wavelength spectra. $Results$. The TeV electron spectra and magnetic field strengths are consistent within the uncertainties for the three regions, with energy densities of the order 10$^{−12}$ erg cm$^{−3}$. The data indicate the presence of a cutoff in the electron spectrum at energies of $\sim$100 TeV and a magnetic field strength of $\sim$6 $\mu$G. Constraints on the presence of turbulent magnetic fields are weak. $Conclusions$. The pressure of TeV electrons and magnetic fields in the cocoon is dynamically negligible, requiring the presence of another dominant pressure component to balance the pulsar wind at the termination shock. Sub-TeV electrons cannot completely account for the missing pressure, which may be provided either by relativistic ions or from mixing of the ejecta with the pulsar wind. The electron spectra are consistent with expectations from transport scenarios dominated either by advection via the reverse shock or by diffusion, but for the latter the role of radiative losses near the termination shock needs to be further investigated in the light of the measured cutoff energies. Constraints on turbulent magnetic fields and the shape of the electron cutoff can be improved by spectral measurements in the energy range ≳ 10 keV.< Leer menos