We present a comprehensive analysis of all XMM-Newton spectra of OJ 287 spanning 15 yr of X-ray spectroscopy of this bright blazar. We also report the latest results from our dedicated Swift UVOT and XRT monitoring of OJ 287, which started in 2015, along with all earlier public Swift data since 2005. During this time interval, OJ 287 was caught in extreme minima and outburst states. Its X-ray spectrum is highly variable and encompasses all states seen in blazars from very flat to exceptionally steep. The spectrum can be decomposed into three spectral components: Inverse Compton (IC) emission dominant at low-state, supersoft synchrotron emission that becomes increasingly dominant as OJ 287 brightens, and an intermediately-soft (Γx = 2.2) additional component seen at outburst. This last component extends beyond 10 keV and plausibly represents either a second synchrotron/IC component and/or a temporary disc corona of the primary supermassive black hole (SMBH). Our 2018 XMM-Newton observation, quasi-simultaneous with the Event Horizon Telescope observation of OJ 287, is well described by a two-component model with a hard IC component of Γx = 1.5 and a soft synchrotron component. Low-state spectra limit any long-lived accretion disc/corona contribution in X-rays to a very low value of Lx/LEdd < 5.6 × 10-4 (for MBH, primary = 1.8 × 1010 M⊙). Some implications for the binary SMBH model of OJ 287 are discussed.