GRB 120323A is a very intense short Gamma Ray Burst (GRB) detected simultaneously during its prompt gamma-ray emission phase with the Gamma-ray Burst Monitor (GBM) on board the Fermi Gamma-ray Space Telescope and the Konus experiment on board the Wind satellite. GBM and Konus operate in the keV--MeV regime, however, the GBM range is broader both toward the low and the high parts of the gamma-ray spectrum. Analysis of such bright events provide a unique opportunity to check the consistency of the data analysis as well as cross-calibrate the two instruments. We performed time-integrated and coarse time-resolved spectral analysis of GRB 120323A prompt emission. We conclude that the analyses of GBM and Konus data are only consistent when using a double-hump spectral shape for both data sets; in contrast, the single-hump of the empirical Band function, traditionally used to fit GRB prompt emission spectra, leads to significant discrepancies between GBM and Konus analysis results. Our two-hump model is a combination of a thermal-like and a non-thermal component. We interpret the first component as a natural manifestation of the jet photospheric emission.