  This paper is the third one in a series devoted to studying the properties of
binaries involving M giants. We use a new set of orbits to construct the first
(e-logP) diagram of an extensive sample of M giant binaries, to obtain their
mass-function distribution, and to derive evolutionary constraints for this
class of binaries and related systems. The orbital properties of binaries
involving M giants were analysed and compared with those of related families of
binaries (K giants, post-AGB stars, barium stars, Tc-poor S stars). The orbital
elements of post-AGB stars and M giants are not different, which may very
indicate that, for the considered sample of post-AGB binaries, the post-AGB
star left the AGB at quite an early stage (M4 or so). Neither are the orbital
elements of post-mass-transfer binaries like barium stars very different from
those of M giants, suggesting that the mass transfer did not alter the orbital
elements much, contrary to current belief. Finally, we show that binary systems
with e < 0.4 log P - 1 (with periods expressed in days) are predominantly
post-mass-transfer systems, because (i) the vast majority of barium and S
systems match this condition, and (ii) these systems have companion masses
peaking around 0.6 solar mass, as expected for white dwarfs. The latter
property has been shown to hold as well for open-cluster binaries involving K
giants, for which a lower bound on the companion mass may easily be set.
