  Electrons carrying orbital angular momentum (OAM) have recently been
discovered theoretically and obtained experimentally that opens up
possibilities for using them in high-energy physics. We consider such a twisted
electron moving in external field of a plane electromagnetic wave and study how
this field influences the electron's OAM. Being motivated by the development of
high-power lasers, we focus our attention on a classically strong field regime
for which $-e^2 \bar {A^2}/m_e^2 c^4 \gtrsim 1$. It is shown that along with
the well-known "plane-wave" Volkov solution, Dirac equation also has the
"non-plane-wave" solutions, which possess OAM and a spin-orbit coupling, and
generalize the free-electron's Bessel states. Motion of the electron with OAM
in a circularly polarized laser wave reveals a twofold character: the
wave-packet center moves along a classical helical trajectory with some quantum
transverse broadening (due to OAM) existing even for a free electron. Using the
twisted states, we calculate the electron's total angular momentum and predict
its shift in the strong-field regime that is analogous to the well-known shifts
of the electron's momentum and mass (and to a less known shift of its spin) in
intense fields. Since the electron's effective angular momentum is conserved in
a plane wave, as well as in some more general field configurations, we discuss
several possibilities for accelerating non-relativistic twisted electrons by
using the focused and combined electromagnetic fields.
