  We report on the discovery of four millisecond pulsars (MSPs) in the High
Time Resolution Universe (HTRU) pulsar survey being conducted at the Parkes
64-m radio telescope. All four MSPs are in binary systems and are likely to
have white dwarf companions. In addition, we present updated timing solutions
for 12 previously published HTRU MSPs, revealing new observational parameters
such as five proper motion measurements and significant temporal dispersion
measure variations in PSR J1017-7156. We discuss the case of PSR J1801-3210,
which shows no significant period derivative after four years of timing data.
Our best-fit solution shows a period derivative of the order of $10^{-23}$, an
extremely small number compared to that of a typical MSP. However, it is likely
that the pulsar lies beyond the Galactic Centre, and an unremarkable intrinsic
period derivative is reduced to close to zero by the Galactic potential
acceleration. Furthermore, we highlight the potential to employ PSR J1801-3210
in the strong equivalence principle test due to its wide and circular orbit. In
a broader comparison with the known MSP population, we suggest a correlation
between higher mass functions and the presence of eclipses in `very low-mass
binary pulsars', implying that eclipses are observed in systems with high
orbital inclinations. We also suggest that the distribution of the total mass
of binary systems is inversely-related to the Galactic height distribution.
Finally, we report on the first detection of PSRs J1543-5149 and J1811-2404 as
gamma-ray pulsars.
