  The probability to record a twisted photon produced by a cold relativistic
particle bunch of charged particles is derived. The radiation of twisted
photons by such particle bunches in stationary electromagnetic fields and in
propagating electromagnetic waves is investigated. It is shown that the
incoherent radiation by bunches of particles traversing normally an isotropic
dispersive medium (the edge, transition, and Vavilov-Cherenkov radiations) and
by bunches moving in a helical undulator does not depend on the azimuthal
distribution of particles in the bunch and is the same as for round bunches. As
for planar undulators, the incoherent radiation by particle bunches is the same
as for the bunches symmetric under reflection with respect to the axis of a
twisted photon detector. At high energies of recorded twisted photons, this
property is universal and holds for the forward incoherent radiation by any
cold relativistic particle bunch. The coherent radiation of twisted photons by
such particle bunches obeys the property that we call the addition rule. This
rule provides a simple means to describe the properties of coherent radiation
of twisted photons. Furthermore, the strong addition rule is established for
the coherent radiation by sufficiently long helical bunches. The use of this
rule allows one to elaborate superradiant pure sources of twisted photons. The
coherent radiation by helical bunches is considered for the edge, transition,
and Vavilov-Cherenkov processes and for particles moving in undulators and
plane laser waves with circular polarization. In these cases, the sum rules are
deduced for the total probability to record a twisted photon and for the
projection of the total angular momentum per photon. The explicit expressions
for both incoherent and coherent interference factors are derived for several
simple bunch profiles.
