  Reconsideration of the T-duality of the open string allows us to introduce
some geometric features in non-geometric theories. First, we have found what
symmetry is T-dual to the local gauge transformations. It includes
transformations of background fields but does not include transformations of
the coordinates. According to this we have introduced a new, up to now missing
term, with additional gauge field $A^D_i$ (D denotes components with Dirichlet
boundary conditions). It compensates non-fulfilment of the invariance under
such transformations on the end-points of an open string, and the standard
gauge field $A^N_a$ (N denotes components with Neumann boundary conditions)
compensates non-fulfilment of the gauge invariance. Using a generalized
procedure we will perform T-duality of vector fields linear in coordinates. We
show that gauge fields $A^N_a$ and $A^D_i$ are T-dual to ${}^\star A_D^a$ and
${}^\star A_N^i$ respectively.
  We introduce the field strength of T-dual non-geometric theories as
derivatives of T-dual gauge fields along both T-dual variable $y_\mu$ and its
double ${\tilde y}_\mu$. This definition allows us to obtain gauge
transformation of non-geometric theories which leaves the T-dual field strength
invariant. Therefore, we introduce some new features of non-geometric theories
where field strength has both antisymmetric and symmetric parts. This allows us
to define new kinds of truly non-geometric theories.
