  A new version of PT-symmetric quantum theory is proposed and illustrated by
an N-site-lattice Legendre oscillator. The essence of the innovation lies in
the replacement of parity P (serving as an indefinite metric in an auxiliary
Krein space) by its non-involutory alternative P(positive)=Q>0 playing the role
of a positive-definite nontrivial metric in an auxiliary, redundant, unphysical
Hilbert space. It is shown that the QT-symmetry of this form remains appealing
and technically useful.
