  LISA will detect $\sim \! 10^4$ Galactic binaries, the majority being double
white dwarfs. However, approximately $\sim \! 1 \textrm{--} 5 \%$ of these
systems will contain neutron stars which, if they can be correctly identified,
will provide new opportunities for studying binary evolution pathways involving
mass reversal and supernovae as well as being promising targets for
multi-messenger observations. Eccentricity, expected from neutron star natal
kicks, will be a key identifying signature for binaries containing a neutron
star. Eccentric binaries radiate at widely-spaced frequency harmonics that must
first be identified as originating from a single source and then analysed
coherently. A multi-harmonic heterodyning approach for this type of data
analysis is used to perform Bayesian parameter estimation on a range of
simulated eccentric LISA signals. This is used to: (i) investigate LISA's
ability to measure orbital eccentricity and to quantify the minimum detectable
eccentricity; (ii) demonstrate how eccentricity and periastron precession help
to break the mass degeneracy allowing the individual component masses to be
inferred, potentially confirming the presence of a neutron star; (iii)
investigate the possibility of source misidentification when the individual
harmonics of an eccentric binary masquerade as separate circular binaries; and
(iv) investigate the possibility of source reclassification, where parameter
estimation results of multiple circular analyses are combined in postprocessing
to quickly infer the parameters of an eccentric source. The broader
implications of this for the ongoing design of the LISA global fit are also
discussed.
