Measuring the abundances of C- and O-bearing species in exoplanet atmospheres enables us to constrain the C/O ratio, which contains indications about the planet formation history. With a wavelength coverage going from 0.95 to 2.5 µm, the high-resolution (R ~ 70 000) spectropolarimeter SPIRou can detect spectral lines of major bearers of C and O in exoplanets. Here, we present our study of SPIRou transmission spectra of WASP-76 b acquired for the ATMOSPHERIX programme. We applied the publicly available data analysis pipeline developed within the ATMOSPHERIX Consortium, analysing the data using 1-D models created with the petitRADTRANS code, with and without a grey cloud deck. We report the detection of H2O and CO at a Doppler shift of around −6 km s−1, which is consistent with previous observations of the planet. Finding that a deep cloud deck is favoured, we measured a mass mixing ratio (MMR) log(H2O)MMR = −4.52 ± 0.77 and log(CO)MMR = −3.09 ± 1.05 consistent with a sub-solar metallicity to more than 1σ. We report 3σ upper limits for the abundances of C2H2, HCN, and OH. We estimated a C/O ratio of 0.94 ± 0.39 (~ 1.7 ± 0.7 × solar, with errors indicated corresponding to the 2σ values) for the limbs of WASP-76 b at the pressures probed by SPIRou. We used 1-D ATMO forward models to verify the validity of our estimation. Comparing them to our abundance estimations of H2O and CO, as well as our upper limits for C2H2, HCN and OH, we found that our results were consistent with a C/O ratio between 1 and 2 × solar, and hence with our C/O estimation. Finally, we found indications of asymmetry for both H2O and CO when investigating the dynamics of their signatures, pointing to a complex scenario possibly involving both a temperature difference between limbs and the presence of clouds being behind the asymmetry that this planet is best known for.