  In this article we offer the new interpretation of T-dualization procedure of
type II superstring theory in double space framework. We use the ghost free
action of type II superstring in pure spinor formulation in approximation of
constant background fields up to the quadratic terms. T-dualization along any
subset of the initial coordinates, $x^a$, is equivalent to the permutation of
this subset with subset of the corresponding T-dual coordinates, $y_a$, in
double space coordinate $Z^M=(x^\mu,y_\mu)$. Demanding that the T-dual
transformation law after exchange $x^a\leftrightarrow y_a$ has the same form as
initial one, we obtain the T-dual NS-NS and NS-R background fields. The T-dual
R-R field strength is determined up to one arbitrary constant under some
assumptions. The compatibility between supersymmetry and T-duality produces
change of bar spinors and R-R field strength. If we dualize odd number of
dimensions $x^a$, such change flips type IIA/B to type II B/A. If we T-dualize
time-like direction, one imaginary unit $i$ maps type II superstring theories
to type $II^\star$ ones.
