  We show that the quadratic transportation cost inequality $T_2$ is equivalent
to both a Poincar\'e inequality and a strong form of the Gaussian concentration
property. The main ingredient in the proof is a new family of inequalities,
called modified quadratic transportation cost inequalities in the spirit of the
modified logarithmic-Sobolev inequalities by Bobkov and Ledoux \cite{BL97},
that are shown to hold as soon as a Poincar\'e inequality is satisfied.
