  The usual Weyl calculus is intimately associated with the choice of the
standard symplectic structure on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. In this
paper we will show that the replacement of this structure by an arbitrary
symplectic structure leads to a pseudo-differential calculus of operators
acting on functions or distributions defined, not on $\mathbb{R}^{n}$ but
rather on $\mathbb{R}^{n}\oplus\mathbb{R}^{n}$. These operators are intertwined
with the standard Weyl pseudo-differential operators using an infinite family
of partial isometries of $L^{2}(\mathbb{R}^{n})\longrightarrow
L^{2}(\mathbb{R}^{2n})$ \ indexed by $\mathcal{S}(\mathbb{R}^{n})$. This allows
us obtain spectral and regularity results for our operators using Shubin's
symbol classes and Feichtinger's modulation spaces.
