Detailed spectroscopy of the neutron-deficient nucleus 36Ca was obtained up to 9 MeV using the 37Ca(p,d)36Ca and the 38Ca(p,t)36Ca transfer reactions. The radioactive nuclei, produced by the LISE spectrometer at GANIL, interacted with the protons of the liquid Hydrogen target CRYPTA, to produce light ejectiles (the deuteron d or triton t) that were detected in the MUST2 detector array, in coincidence with the heavy residues %identified by a zero degree detection system. %States have been measured up to 9 MeV. Our main findings are: i) a similar shift in energy for the 11+ and 21+ states by about -250 keV, as compared to the mirror nucleus 36S, ii) the discovery of an intruder 02+ state at 2.83(13) MeV, which appears below the first 2+ state, in contradiction with the situation in 36S, and iii) a tentative 03+ state at 4.83(17) MeV, proposed to exhibit a bubble structure with two neutron vacancies in the 2s1/2 orbit. The inversion between the 02+ and 21+ states is due to the large mirror energy difference (MED) of -516(130) keV for the former. This feature is reproduced by Shell Model (SM) calculations, using the sd-pf valence space, predicting an almost pure intruder nature for the 02+ state, with two protons (neutrons) being excited across the Z=20 magic closure in 36Ca (36S). This mirror system has the largest MEDs ever observed, if one excludes the few cases induced by the effect of the continuum