  We have subjected the planar pendulum eigenproblem to a symmetry analysis
with the goal of explaining the relationship between its conditional
quasi-exact solvability (C-QES) and the topology of its eigenenergy surfaces,
established in our earlier work [Frontiers in Physical Chemistry and Chemical
Physics 2, 1-16, (2014)]. The present analysis revealed that this relationship
can be traced to the structure of the tridiagonal matrices representing the
symmetry-adapted pendular Hamiltonian, as well as enabled us to identify many
more -- forty in total to be exact -- analytic solutions. Furthermore, an
analogous analysis of the hyperbolic counterpart of the planar pendulum, the
Razavy problem, which was shown to be also C-QES [American Journal of Physics
48, 285 (1980)], confirmed that it is anti-isospectral with the pendular
eigenproblem. Of key importance for both eigenproblems proved to be the
topological index $\kappa$, as it determines the loci of the intersections
(genuine and avoided) of the eigenenergy surfaces spanned by the dimensionless
interaction parameters $\eta$ and $\zeta$. It also encapsulates the conditions
under which analytic solutions to the two eigenproblems obtain and provides the
number of analytic solutions. At a given $\kappa$, the anti-isospectrality
occurs for single states only (i.e., not for doublets), like C-QES holds solely
for integer values of $\kappa$, and only occurs for the lowest eigenvalues of
the pendular and Razavy Hamiltonians, with the order of the eigenvalues
reversed for the latter. For all other states, the pendular and Razavy spectra
become in fact qualitatively different, as higher pendular states appear as
doublets whereas all higher Razavy states are singlets.
