The highest-temperature superconductors are electronically inhomogeneous at the nanoscale, suggesting the existence of a local variable that could be harnessed to enhance the superconducting pairing. Here we report the relationship between local doping and local strain in the cuprate superconductor Bi2Sr2CaCu2O8+x. We use scanning tunneling microscopy to discover that the crucial oxygen dopants are periodically distributed in correlation with local strain. Our picoscale investigation of the intraunit-cell positions of all oxygen dopants provides essential structural input for a complete microscopic theory.