  The symmetry operations of the crystal groups relevant for the high
temperature superconductors HgBa2CuO4+x (Hg1201), YBa2Cu3O7-x (YBCO), and
  Bi2Sr2CaCu2O8+x (BSCCO) are elucidated. The allowable combinations of the
superconducting order parameter (OP) components are presented for both the
angular momentum and lattice representations. For tetragonal Hg1201, the spin
singlet OP components are composed from four sets of compatible basis
functions, which combine to give the generalized s-, dx2-y2-, dxy-, and
gxy(x2-y2)- wave OPs. In YBCO, elements of s- and dx2-y2- wave sets are
compatible, but in BSCCO, elements of s- and dxy- wave sets are compatible. The
Josephson critical current density JcJ across c-axis twist junctions in the
vicinity of Tc is then evaluated as a function of the twist angle phi0, for
each allowable OP combination, for both coherent and incoherent tunneling.
Recent experiments of Li et al. demonstrated the independence of JcJ(phi0)/JcS
upon phi0 at and below Tc, where JcS is the critical current density of a
constituent single crystal. These experiments are shown to be consistent with
an OP containing an s-wave component, but inconsistent with an OP containing
the purported dx2-y2-wave component. In addition, they demonstrate that the
interlayer tunneling across untwisted layers in single crystal BSCCO is
entirely incoherent. We propose a new type of tricrystal experiment using
single crystal c-axis twist junctions, that does not employ substrate grain
boundaries.
