Recent analysis of the Homestake data has yielded evidence that the solar n
eutrino flux varies in time-more specifically, that it exhibits a periodic
variation that may be attributed to rotational modulation occurring deep in
the solar interior, either in the tachocline or in the radiative zone. Her
e we present a spectral analysis of the GALLEX data that yields supporting
evidence for this rotational modulation. The most prominent peak in the pow
er spectrum occurs at the synodic frequency of 13.08 yr(-1) (cycles per yea
r) and is estimated to be significant at the 0.1% level. It appears that th
e most likely interpretation of this modulation is that the electron neutri
nos have nonzero magnetic moment, so that they oscillate between left-hand
(detectable) and right-hand (nondetectable) chiralities as they traverse th
e Sun's internal magnetic field. This oscillation could account for the neu
trino deficit. The second strongest peak in the GALLEX spectrum has a perio
d of 52 days, and this period occurs in other solar data as well. We sugges
t that this periodicity and also the Rieger 154 day periodicity, which show
s up in many solar parameters and in the Homestake data, are due to r-mode
oscillations.