  The general spin structure of the matrix element, taking into account the
two--photon exchange contribution, for the elastic electron (positron)
--deuteron scattering has been derived using general symmetry properties of the
hadron electromagnetic interaction, such as P--, C-- and T--invariances as well
as lepton helicity conservation in QED at high energy. Taking into account also
crossing symmetry, the amplitudes of $e^{\mp}d-$scattering can be parametrized
in terms of fifteen real functions. The expressions for the differential cross
section and for all polarization observables are given in terms of these
functions. We consider the case of an arbitrary polarized deuteron target and
polarized electron beam (both longitudinal and transverse). The transverse
polarization of the electron beam induces a single--spin asymmetry which is
non--zero in presence of two--photon exchange. It is shown that elastic
deuteron electromagnetic form factors can still be extracted in presence of two
photon exchange, from the measurements of the differential cross section and of
one polarization observable (for example, the tensor asymmetry) for electron
and positron deuteron elastic scattering, in the same kinematical conditions.
