  We consider the application of a basic principle of quantum theory, the
tracelessness of a certain class of hamiltonians, to the precision spectroscopy
of the molecular hydrogen ions. We show that it is possible to obtain the
spin-averaged transition frequencies between states from a simple weighted sum
of experimentally accessible spin-dependent transition frequencies. We discuss
the cases ${\rm H}_{2}^{+}$ and ${\rm HD}^{+}$, which are distinct in the
multipole character of their rovibrational transitions. Inclusion of additional
frequencies permits canceling also the electric quadrupole shift, the Zeeman
shift and partially the Stark shift. In this context, we find that measuring
electric quadrupole transitions in ${\rm HD}^{+}$ is advantageous. The required
experimental effort appears reasonable.
