  For a twisted (vortex) Dirac particle in nonuniform electric and magnetic
fields, the relativistic Foldy-Wouthuysen Hamiltonian is derived including high
order terms describing new effects. The result obtained shows for the first
time that a twisted spin-1/2 particle possesses a tensor magnetic
polarizability and a measurable (spectroscopic) electric quadrupole moment. We
have calculated the former parameter and have evaluated the latter one for a
twisted electron. The tensor magnetic polarizability of the twisted electron
can be measured in a magnetic storage ring because a beam with an initial
orbital tensor polarization acquires a horizontal orbital vector polarization.
The electric quadrupole moment is rather large and strongly influences the
dynamics of the intrinsic orbital angular momentum. Three different methods of
its measurements, freezing the intrinsic orbital angular momentum and two
resonance methods, are proposed. The existence of the quadrupole moment of
twisted electrons can lead to practical applications.
