  Some of the most exotic material properties derive from electronic states
with short correlation length (~10-500 {\AA}), suggesting that the local
structural symmetry may play a relevant role in their behavior. Here we discuss
the combined use of polarized x-ray absorption fine structure and x-ray
diffraction at high pressure as a powerful method to tune and probe structural
and electronic orders at multiple length scales. Besides addressing some of the
technical challenges associated with such experiments, we illustrate this
approach with results obtained in the cuprate La$_{1.875}$Ba$_{0.125}$CuO$_4$,
in which the response of electronic order to pressure can only be understood by
probing the structure at the relevant length scales.
