  We show that reflection symmetry has a strong influence on quantum transport
properties. Using a random S-matrix theory approach, we derive the
weak-localization correction, the magnitude of the conductance fluctuations,
and the distribution of the conductance for three classes of reflection
symmetry relevant for experimental ballistic microstructures. The S-matrix
ensembles used fall within the general classification scheme introduced by
Dyson, but because the conductance couples blocks of the S-matrix of different
parity, the resulting conductance properties are highly non-trivial.
