Using data collected with the BeppoSAX, INTEGRAL and Swift satellites, we report and discuss the results of a study on the X-ray emission properties of the X-ray source 1ES 1210-646, recently classified as a high-mass X-ray binary through optical spectroscopy. This is the first in-depth analysis of the X-ray spectral characteristics of this source. We found that the flux of 1ES 1210-646 varies by a factor of about 3 on a timescale of hundreds of seconds and by a factor of at least 10 among observations acquired over a time span of several months. The X-ray spectrum of 1ES 1210-646 is described using a simple powerlaw shape or, in the case of INTEGRAL data, with a blackbody plus powerlaw model. Spectral variability is found in connection with different flux levels of the source. A strong and transient iron emission line with an energy of about 6.7 keV and an equivalent width of about 1.6 keV is detected when the source is found at an intermediate flux level. The line strength seems to be tied to the orbital motion of the accreting object, as this feature is only apparent at the periastron. Although the X-ray spectral description we find for the 1ES 1210-646 emission is quite atypical for a high-mass X-ray binary, the multiwavelegth information available for this object leads us to confirm this classification. The results presented here allow us instead to definitely rule out the possibility that 1ES 1210-646 is a (magnetic) cataclysmic variable as proposed previously and, in a broader sense, a white dwarf nature for the accretor is disfavoured. X-ray spectroscopic data actually suggest a neutron star with a low magnetic field as the accreting object in this system. Comment: 10 pages, 5 figures, 3 tables, accepted for publication on Astronomy and Astrophysics, main journal