We report on the design and performance of a double-sided coincidence velocity map imaging spectrometer optimized for electron-ion and ion-ion coincidence experiments studying inner-shell photoionization of gas-phase molecules with soft X-ray synchrotron radiation. The apparatus employs two microchannel plate detectors equipped with delay-line anodes for coincident, time- and position-resolved detection of photoelectrons and Auger electrons with kinetic energies up to 300 eV on one side of the spectrometer and photoions up to 25 eV per unit charge on the opposite side. We demonstrate its capabilities by measuring valence photoelectrons and ion spectra of neon and nitrogen and by studying channel-resolved photoelectron and Auger spectra along with fragment-ion momentum correlations for chlorine 2p inner-shell ionization of cis- and trans-1,2-dichloroethene.