  In this paper we present a novel quasi-exactly solvable model with symmetric
inverted potentials which are unbounded from below. The quasi-exactly solvable
states are shown to be total transmission (or reflectionless) modes. From these
modes even and odd wavefunctions can be constructed which are normalizable and
flux-zero. Under the procedure of self-adjoint extension, a discrete spectrum
of bound states can be obtained for these inverted potentials and the solvable
part of the spectrum is the quasi-exactly solvable states we have discovered.
