  We study the capture and subsequent annihilation of inelastic dark matter
(DM) in the Sun, placing constraints on the DM-nucleon scattering cross section
from the null result of the IceCube neutrino telescope. We then compare such
constraints with exclusion limits on the same cross section that we derive from
XENON1T, PICO and CRESST results. We calculate the cross section for inelastic
DM-nucleon scattering within an extension of the effective theory of DM-nucleon
interactions which applies to the case of inelastic DM models characterised by
a mass splitting between the incoming and outgoing DM particle. We find that
for values of the mass splitting parameter larger than about 200 keV, neutrino
telescopes place limits on the DM-nucleon scattering cross section which are
stronger than the ones from current DM direct detection experiments. The exact
mass splitting value for which this occurs depends on whether DM thermalises in
the Sun or not. This result applies to all DM-nucleon interactions that
generate DM-nucleus scattering cross sections which are independent of the
nuclear spin, including the "canonical" spin-independent interaction. We
explicitly perform our calculations for a DM candidate with mass of 1 TeV, but
our conclusions qualitatively also apply to different masses. Furthermore, we
find that exclusion limits from IceCube on the coupling constants of this
family of spin-independent interactions are more stringent than the ones from a
(hypothetical) reanalysis of XENON1T data based on an extended signal region in
nuclear recoil energy. Our results should be taken into account in global
analyses of inelastic DM models.
