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An accessible tabletop source for the production of entangled x rays is crucial for the field of high-energy quantum optics. Here, we present a detailed analysis of the entanglement and polarization of the two photons emitted by an electron in an intense laser wave (nonlinear double-Compton scattering), by working within the framework of strong-field QED. By identifying contributions to the emission probability stemming from the electron being either on-shell or off-shell between the two-photon emissions, we show that the entangled photons are generated via the off-shell contribution for moderate beam energies, which are realizable on a tabletop setup. We propose an experiment to produce and isolate pairs of entangled x rays, through spectral filtering.