  The Bianchi IX mixmaster model is quantized in its non-diagonal form,
imposing spatial diffeomorphism, time reparametrization and Lorentz invariance
as constraints on physical state vectors before gauge-fixing. The result turns
out to be different from quantizing the diagonal model obtained by gauge-fixing
already on the classical level. For the non-diagonal model a generalized
9-dimensional Fourier transformation over a suitably chosen manifold connects
the representations in metric variables and in Ashtekar variables. A space of
five states in the metric representation is generated from the single physical
Chern-Simons state in Ashtekar variables by choosing five different integration
manifolds, which cannot be deformed into each other. For the case of a positive
cosmological constant $\Lambda$ we extend our previous study of these five
states for the diagonal Bianchi IX model to the non-diagonal case. It is shown
that additional discrete (permutation) symmetries of physical states arise in
the quantization of the non-diagonal model, which are satisfied by two of the
five states connected to the Chern-Simons state. These have the characteristics
of a wormhole groundstate and a Hartle-Hawking `no-boundary' state,
respectively. We also exhibit a special gauge-fixing of the time
reparametrization invariance of the quantized system and define an associated
manifestly positive scalar product. Then the wormhole ground state is left as
the only normalizable physical state connected to the Chern-Simons state.
