We study global reflection symmetries of almost periodic functions. In the
non-limit periodic case, we establish an upper bound on the Haar measure of
the set of those elements in the hull which are almost symmetric about the
origin. As an application of this result we prove that in the non-limit pe
riodic case, the criterion of Jitomirskaya and Simon ensuring absence of ei
genvalues for almost periodic Schrodinger operators is only applicable on a
set of zero Haar measure. We complement this by giving examples of limit p
eriodic functions where the Jitomirskaya Simon criterion can be applied to
every element of the hull. (C) 2000 Academic Press.