  In this paper we characterize global regularity in the sense of Shubin of
twisted partial differential operators of second order in dimension $2$. These
operators form a class containing the twisted Laplacian, and in bi-unique
correspondence with second order ordinary differential operators with
polynomial coefficients and symbol of degree $2$. This correspondence is
established by a transformation of Wigner type. In this way the global
regularity of twisted partial differential operators turns out to be equivalent
to global regularity and injectivity of the corresponding ordinary differential
operators, which can be completely characterized in terms of the asymptotic
behavior of the Weyl symbol. In conclusion we observe that we have obtained a
new class of globally regular partial differential operators which is disjoint
from the class of hypo-elliptic operators in the sense of Shubin.
