  The hard X-ray twisted photons and relativistic massive particles with
orbital angular momentum -- vortex electrons, muons, protons, etc. -- have many
potential applications in high-energy and nuclear physics. However, such states
can be obtained so far mainly via diffraction techniques, not applicable for
relativistic energies. Here we show that the vortex states of different
particles, including hadrons, ions, and nuclei, can be generated in a large
class of processes with two final particles simply by altering a postselection
protocol. Thanks to entanglement and to the uncertainty relations, an evolved
state of a final particle becomes twisted if the momentum azimuthal angle of
the other particle is measured with a large uncertainty. We give several
examples, including Cherenkov and undulator radiation, particle collisions with
intense laser beams, $e\mu \to e\mu, ep \to ep$. This technique can be adapted
for ultrarelativistic lepton and hadron beams of linear colliders, and it can
also facilitate the development of sources of X-ray and $\gamma$-range twisted
photons at storage rings and free-electron lasers.
