  We study spontaneous emission of a photon during the transitions between
relativistic Landau states of an electron in a constant magnetic field that can
reach the Schwinger value of $H_c = 4.4 \times 10^9$ T. In contrast to the
conventional method in which detection of both the final electron and the
photon is implied in a certain basis, here we derive the photon state as it
evolves from the process itself. It is shown that the emitted photon state
represents a twisted Bessel beam propagating along the field axis with a total
angular momentum (TAM) projection onto this axis $\ell-\ell'$ where $\ell$ and
$\ell'$ are the TAM of the initial electron and of the final one, respectively.
Thus, the majority of the emitted photons turn out to be twisted with
$\ell-\ell' \gtrsim 1$, even when the magnetic field reaches the critical value
of $H\sim H_c$. The transitions without a change of the electron angular
momentum, $\ell'=\ell$, are possible, yet much less probable. We also compare
our findings with those for a spinless charged particle and demonstrate their
good agreement for the transitions without change of the electron spin
projection even in the critical fields, while the spin-flip transitions are
generally suppressed. In addition, we argue that whereas the ambiguous choice
of an electron spin operator affects the differential probability of emission,
this problem can partially be circumvented for the photon evolved state because
it is the electron TAM rather than the spin alone that defines the TAM of the
emitted twisted photon.
