  We study the points of degeneracy (diabolical points) in magnetic molecules
such as Mn_{12}-acetate that have an easy axis of four-fold symmetry. This is
done for general magnetic field that need not be oriented along a high-symmetry
direction. We develop a perturbative technique that gives the diabolical points
as the roots of a small number of polynomials in the transverse component of
the magnetic field and the fourth order basal plane anisotropy. In terms of
these roots we obtain approximate analytic formulas that apply to any system
with total spin S \le 10. The analytic results are found to compare reasonably
well with exact numerical diagonalization for the case of Mn12. In addition,
the perturbation theory shows that the diabolical points may be indexed by the
magnetic quantum numbers of the levels involved, even at large transverse
fields. Certain points of degeneracy are found to be mergers (or near mergers)
of two or three diabolical points beacuse of the symmetry of the problem.
