We report on the X-ray and optical properties of two high-mass X-ray binary systems located in the Large Magellanic Cloud (LMC). Based on the obtained optical spectra, we classify the massive companion as a supergiant star in both systems. Timing analysis of the X-ray events collected by XMM-Newton revealed the presence of coherent pulsations (spin period ∼2013 s) for XMMU J053108.3-690923 and fast flaring behaviour for XMMU J053320.8-684122. The X-ray spectra of both systems can be modelled sufficiently well by an absorbed power law, yielding hard spectra and high intrinsic absorption from the environment of the systems. Due to their combined X-ray and optical properties, we classify both systems as SgXRBs: the 19th confirmed X-ray pulsar and a probable supergiant fast X-ray transient in the LMC, the second such candidate outside our Galaxy.