  We apply the semiclassical theory of transport to quantum dots with exact and
approximate spatial symmetries; left-right mirror symmetry, up-down mirror
symmetry, inversion symmetry or four-fold symmetry. In this work - the first of
a pair of articles - we consider (a) perfectly symmetric dots and (b) nearly
symmetric dots in which the symmetry is broken by the dot's internal dynamics.
The second article addresses symmetry-breaking by displacement of the leads.
Using semiclassics, we identify the origin of the symmetry-induced interference
effects that contribute to weak-localization corrections and universal
conductance fluctuations. For perfect spatial symmetry, we recover results
previously found using the random-matrix theory conjecture. We then go on to
show how the results are affected by asymmetries in the dot, magnetic fields
and decoherence. In particular, the symmetry-asymmetry crossover is found to be
described by a universal dependence on an asymmetry parameter $\gamma_{asym}$.
However, the form of this parameter is very different depending on how the dot
is deformed away from spatial symmetry. Symmetry-induced interference effects
are completely destroyed when the dot's boundary is globally deformed by less
than an electron wavelength. In contrast, these effects are only reduced by a
finite amount when a part of the dot's boundary smaller than a lead-width is
deformed an arbitrarily large distance.
