Nature has kindly provided us with a selection of very different star-forming regions in the vicinity of the Sun. While the closely-studied Taurus-Aurigae, Ophiucus, and Lupus star forming regions are low mass forming regions with relatively low stellar density, the Orion star-forming cloud is a dense region in which the effects of environmental influence on forming planetary systems are expected to be largest. Disk imaging in scattered light with ground-based 8m-class telescopes has so far mostly focused on the very nearby low-mass star-forming regions, and revealed a wealth of structure and companions in such objects. We selected a sample of young stars in the Orion cloud and attempt to characterize its (possibly planet-induced) structures and close companions. We present SPHERE/IRDIS H-band data from a sample of 23 stars in the Orion Star forming region observed under the DESTINYS (Disk Evolution Study Through Imaging of Nearby Young Stars) program. The data presented contains 10 disk detections in scattered light out of which three are exceptionally extended. We also present seven detections of double or triple systems out of which five were previously unknown. Among the new detections there are two substellar companions. We use XSHOOTER spectra to characterize the stellar parameters and measure the contrast between the primary and secondary in the double systems to estimate the spectral class and masses of the companions. We compare the scattered light contrast with the age and mass of the stars.