  We study the symmetry classes of graphene quantum dots, both open and closed,
through the conductance and energy level statistics. For abrupt termination of
the lattice, these properties are well described by the standard orthogonal and
unitary ensembles. However, for smooth mass confinement, special time-reversal
symmetries associated with the sublattice and valley degrees of freedom are
critical: they lead to block diagonal Hamiltonians and scattering matrices with
blocks belonging to the unitary symmetry class even at zero magnetic field.
While the effect of this structure is clearly seen in the conductance of open
dots, it is suppressed in the spectral statistics of closed dots, because the
intervalley scattering time is shorter than the time required to resolve a
level spacing in the closed systems but longer than the escape time of the open
systems.
