  Fallback discs around neutron stars (NSs) are believed to be an expected
outcome of supernova explosions. Here we investigate the consequences of such a
common outcome for the timing and spectral properties of the associated NS
population, using Monte Carlo population synthesis models. We find that the
long-term torque exerted by the fallback disc can substantially influence the
late-time period distribution, but with quantitative differences which depend
on whether the initial spin distribution is dominated by slow or fast pulsars.
For the latter, a single-peaked initial spin distribution becomes bimodal at
later times. Timing ages tend to underestimate the real age of older pulsars,
and overestimate the age of younger ones. Braking indices cluster in the range
1.5 <~ n <~ 3 for slow-born pulsars, and -0.5 <~ n <~ 5 for fast-born pulsars,
with the younger objects found predominantly below n <~ 3. Large values of n,
while not common, are possible, and associated with torque transitions in the
NS+disc system. The 0.1-10 keV thermal luminosity of the NS+disc system is
found to be generally dominated by the disc emission at early times, t <~ 10^3
yr, but this declines faster than the thermal surface emission of the NS.
Depending on the initial parameters, there can be occasional periods in which
some NSs switch from the propeller to the accretion phase, increasing their
luminosity up to the Eddington limit for ~ 10^3-10^4 years.
