We present the results of the simultaneous deep XMM-Newton and Chandra observations of the bright Seyfert 1.9 galaxy MCG -5-23-16, which is thought to have one of the best known examples of a relativistically broadened iron Kα line. The time-averaged spectral analysis shows that the iron K-shell complex is best modeled with an unresolved narrow emission component (FWHM<5000 km s-1, EW~60 eV) plus a broad component. This latter component has FWHM~44,000 km s-1 and EW~50 eV. Its profile is well described by an emission line originating from an accretion disk viewed with an inclination angle ~40°, with the emission arising from within a few tens of gravitational radii of the central black hole. The time-resolved spectral analysis of the XMM-Newton EPIC pn spectrum shows that both the narrow and broad components of the Fe K emission line appear to be constant in time within the errors. We detected a narrow sporadic absorption line at 7.7 keV, which appears to be variable on a timescale of 20 ks. If associated with Fe XXVI Lyα, this absorption is indicative of a possibly variable, high-ionization, high-velocity outflow. The variability of this absorption feature appears to rule out a local (z=0) origin. The analysis of the XMM-Newton RGS spectrum reveals that the soft X-ray emission of MCG -5-23-16 is likely dominated by several emission lines superimposed on an unabsorbed scattered power-law continuum. The lack of strong Fe L-shell emission, together with the detection of a strong forbidden line in the O VII triplet, is consistent with a scenario in which the soft X-ray emission lines are produced in a plasma photoionized by the nuclear emission.