  Aims. We aim to investigate how polarimetric observations can improve our
understanding of the nature and diversity of M/X-type asteroids. Methods.
Polarimetric observations of the selected M/X-type asteroids were carried out
at the Tohoku 0.6-m telescope at Haleakala Observatory, Hawaii (simultaneously
in BVR filters), the 2-m telescope of the Bulgarian National Astronomical
Observatory in Rozhen (in R filter), and the 2.15-m telescope of the Complejo
Astron\'omico El Leoncito (CASLEO), Argentina (in V filter). We analysed the
polarimetric characteristics of M/X-type asteroids along with the available
data obtained by other techniques. Results. New polarimetric observations of 22
M/X-type asteroids combined with published observations provide a data set of
41 asteroids for which the depth of a negative polarisation branch and/or
inversion angle were determined. We found that the depth of the negative
polarisation branch tends to increase with decreasing steepness of the
near-infrared spectra. Asteroids with a deeper negative polarisation branch
tend to have a higher radar circular polarisation ratio. We show that, based on
the relationship of the depth of the negative polarisation branch and inversion
angle, two main sub-types can be distinguished among M-type asteroids. We
suggest that these groups may be related to different surface compositions
similar to (1) irons and stony-irons and (2) enstatite and iron-rich
carbonaceous chondrites.
