  We propose of an improved version of the ubiquitous symmetrization inequality
making use of the Wasserstein distance between a measure and its reflection in
order to quantify the symmetry of the given measure. An empirical bound on this
asymmetric correction term is derived through a bootstrap procedure and shown
to give tighter results in practical settings than the original uncorrected
inequality. Lastly, a wide range of applications are detailed including testing
for data symmetry, constructing nonasymptotic high dimensional confidence sets,
bounding the variance of an empirical process, and improving constants in
Nemirovski style inequalities for Banach space valued random variables.
