We present an analysis of the X-ray emission of the rapidly rotating giant star YY Mensae observed by Chandra HETGS and XMM-Newton. The high-resolution spectra display numerous emission lines of highly ionized species; Fe XVII to Fe XXV lines are detected, together with H-like and He-like transitions of lower Z elements. Although no obvious flare was detected, the X-ray luminosity changed by a factor of 2 between the XMM-Newton and Chandra observations taken 4 months apart (from log L_X ≈ 32.2 to 32.5 ergs s_-1, respectively). The coronal abundances and the emission measure distribution have been derived from three different methods using optically thin collisional ionization equilibrium models, which is justified by the absence of opacity effects in YY Men as measured from line ratios of Fe XVII transitions. The abundances show a distinct pattern as a function of the first ionization potential (FIP), suggestive of an inverse FIP effect as seen in several active RS CVn binaries. The low-FIP elements (<10 eV) are depleted relative to the high-FIP elements; when compared to its photospheric abundance, the coronal Fe abundance also appears depleted. We find a high N abundance in YY Men's corona, which we interpret as a signature of material processed in the CNO cycle and dredged up in the giant phase. The corona is dominated by a very high temperature (20-40 MK) plasma, which places YY Men among the magnetically active stars with the hottest coronae. Lower temperature plasma also coexists, albeit with much lower emission measure. Line broadening is reported in some lines, with a particularly strong significance in Ne X Lyα. We interpret such broadening as Doppler thermal broadening, although rotational broadening due to X-ray-emitting material high above the surface could be present as well. We use two different formalisms to discuss the shape of the emission measure distribution. The first one infers the properties of coronal loops, whereas the second formalism uses flares as a statistical ensemble. We find that most of the loops in the corona of YY Men have their maximum temperature equal to or slightly larger than about 30 MK. We also find that small flares could contribute significantly to the coronal heating in YY Men. Although there is no evidence of flare variability in the X-ray light curves, we argue that YY Men's distance and X-ray brightness do not allow us to detect flares with peak luminosities L_X ≤ 10³¹ ergs s^-1 with current detectors.