  The radiation of twisted photons by charged particles traversing a
cholesteric plate is studied in the framework of quantum electrodynamics in an
anisotropic inhomogeneous dispersive medium. The complete set of solutions to
the Maxwell equations in the cholesteric plate is constructed in the paraxial
and small anisotropy approximations. The explicit expressions for the average
numbers of plane-wave and twisted photons created by a charged point particle
crossing the cholesteric plate are derived. The selection rules for the twisted
photons radiated at the harmonic $n\in \mathbb{Z}$ are established. In the
paraxial regime, the projection of the orbital angular momentum of a radiated
twisted photon obeys the selection rule $l=\pm(2n+1)$. In the approximation of
a small anisotropy of the permittivity tensor, the selection rule becomes
$m=\pm 2n$, where $m$ is the projection of the total angular momentum of a
radiated twisted photon. The sign "$\pm$" in these selection rules is related
to the choice of the forward or reflected waves in the cholesteric and is
realized at the different energies of radiated photons. As the examples, the
radiation of optical twisted photons by electrons with the Lorentz factors
$\gamma=235$ and $\gamma=500$ and by uranium nuclei with $\gamma=2$ are
considered. It is shown that charged particles traversing normally a
cholesteric plate can be used as a pure source of twisted photons.
