  Bloch theorem for a periodic operator is being revisited here, and we notice
extra orthogonality relationships. It is shown that solutions are bi-periodic,
in the sense that eigenfunctions are periodic with respect to one argument, and
pseudo-periodic with respect to the other. An additional kind of symmetry
between r-space and k-space exists between the envelope and eignfunctions not
apparently noticed before, which allows to define new invertible modified
Wannier functions. As opposed to the Wannier functions in r-space, these
modified Wannier functions are defined in the k-spaces, but satisfy similar
basic properties. These could result in new algorithms and other novel
applications in the computational tools of periodic structures.
