[en] The physics of photons and electrons carrying orbital angular momentum (OAM) is an exciting field of research in quantum optics and electron microscopy. Usually, one considers propagation of these vortex beams in a medium or an external field and their absorption or scattering on fixed targets. Here we consider instead a beam-beam collision. We show that elastic scattering of a Bessel vortex beam with a counterpropagating plane wave naturally leads to two vortex-entangled outgoing beams. The vortex entanglement implies that the two final particles are entangled not only in their orbital helicities but also in the opening angles of their momentum cones. Our results are driven by kinematics of vortex-beam scattering and apply to particle pairs of any nature: eγ, e+e−, ep, etc. This collisional vortex entanglement can be used to create pairs of OAM-entangled particles of different nature and to transfer a phase vortex, for example, from low-energy electrons to high-energy protons.