  In this paper we use solar neutrino data to derive stringent bounds on
Majorana neutrino transition moments (TMs). Should such be present, they would
contribute to the neutrino--electron scattering cross section and hence alter
the signal observed in Super-Kamiokande. Motivated by the growing robustness of
the LMA-MSW solution of the solar neutrino problem indicated by recent data,
and also by the prospects of its possible confirmation at KamLAND, we assume
the validity of this solution, and we constrain neutrino TMs by using the
latest global solar neutrino data. We find that all elements of the TM matrix
can be bounded at the same time. Furthermore, we show how reactor data play a
complementary role to the solar neutrino data, and use the combination of both
data sets to improve the current bounds. Performing a simultaneous fit of
LMA-MSW oscillation parameters and TMs we find that 6.3 times 10^{-10} mu_B and
2.0 times 10^{-10} mu_B are the 90% C.L. bounds from solar and combined solar +
reactor data, respectively. Finally, we perform a simulation of the upcoming
Borexino experiment and show that it will improve the bounds from today's data
by roughly one order of magnitude.
