  Aims: We calibrate spectroscopically the C- versus (vs.) M-type asymptotic
giant branch (AGB) star selection made using near-IR photometry, and
investigate the spatial distribution of the C/M ratio in NGC 6822, based on low
resolution spectroscopy and near-IR photometry. Methods: We obtained low
resolution multi-object spectroscopy with the VIMOS instrument at the ESO VLT
of ~ 800 stars in seven fields centred on NGC 6822. The spectroscopic
classification of giant stars in NGC 6822 and foreground dwarf contaminants was
made by comparing more than 500 good quality spectra with the spectroscopic
atlas of Turnshek et al. (1985). The sample of spectroscopically confirmed AGB
stars in NGC 6822 is divided into C- and M-rich giants to constrain the C vs. M
AGB star selection criteria based on photometry. The larger near-IR photometric
sample is then used to investigate the C/M ratio gradients across the galaxy.
Results: We present the largest catalogue of near-IR photometry and spectra of
AGB stars in NGC 6822 with 150 C-stars and 122 M-stars. Seventy-nine percent of
the C-stars in our catalogue are redder than (J-K)0=1.2 mag, and 12% are
brighter than K0=16.45 mag and bluer than (J-K)0=1.2 mag. The remaining 9% are
mixed with the M-type AGB stars, 88% of which have colours J-H0>0.73 mag and
(J-K)0 between 0.9 mag and 1.2 mag. The remainder are mixed with dwarfs and
C-type stars. The foreground dwarfs have preferably colours (J-H)0<0.73 mag
(95%). Using the proposed criteria, we estimate that the overall C/M ratio of
the galaxy is around 0.8 with a spread between 0.2<C/M<1.8. These results
suggest that the metallicity index [Fe/H] is between -1.2 dex and -1.3 dex
according to the different calibrations and that there is a significant spread
of about 0.4 - 0.6 dex. We also discuss age rather than metallicity variations
that could explain the C/M ratio trends.
