  We introduce the J map using the Zak transform associated with the Weyl
transform on $L^{2}(\mathbb{R}^{2n})$. We obtain a decomposition for a twisted
shift-invariant subspace of $L^{2}(\mathbb{R}^{2n})$ as a direct sum of
mutually orthogonal principal twisted shift-invariant spaces such that the
respective system of twisted translates forms a Parseval frame sequence. We
establish that the twisted shift preserving operators and the corresponding
range operators simultaneously share some properties in common, namely,
self-adjoint, unitary, range of the spectrum and bounded below properties. We
prove that the frame operator and its inverse associated with a system of
twisted translates of {\varphi_s}_{s\in Z} are shift preserving. We also show
that the corresponding range operators turn out to be the dual Gramian and its
inverse associated with the collection {J\varphi_s(. , .)}_{s\in Z}.
