  We combine experimental constraints from direct detection searches and from
neutrino telescopes looking for WIMP annihilations in the Sun to derive
halo-independent bounds on each of the 28 WIMP-proton and WIMP-neutron
couplings of the effective non-relativistic Hamiltonian that drives the
scattering process off nuclei of a WIMP of spin 1/2. The method assumes that
the velocity distribution is normalized to one and homogeneous at the the solar
system scale, as well as equilibrium between WIMP capture and annihilation in
the Sun, and requires to fix the WIMP annihilation channels (we assume
$b\bar{b}$). We consider a single non-vanishing coupling at a time, and find
that for most of the couplings the degree of relaxation of the halo-independent
bounds compared to those obtained by assuming the Standard Halo Model is with
few exceptions relatively moderate in the low and high WIMP mass regimes, where
it can be as small as a factor of $\simeq 2$, while in the intermediate mass
range between 10 GeV and 200 GeV it can be as large as $\sim 10^3$. An
exception to this general pattern, with more moderate values of the bound
relaxation, is observed in the case of spin-dependent WIMP-proton couplings
with no or a comparatively small momentum suppression, for which WIMP capture
is strongly enhanced because it is driven by scattering events off $^1H$, which
is the most abundant target in the Sun. Within this class of operators the
relaxation is particularly small for interactions that are driven by only the
velocity-dependent term, for which the solar capture signal is enhanced
compared to the direct detection one, thanks to the highest speed of scattering
WIMPs within the Sun due to the larger gravitational acceleration.
