  We study single-ion and exchange anisotropy effects in equal-spin $s_1$
tetramer single molecule magnets exhibiting $T_d$, $D_{4h}$, $D_{2d}$,
$C_{4h}$, $C_{4v}$, or $S_4$ ionic point group symmetry. We first write the
group-invariant quadratic single-ion and symmetric anisotropic exchange
Hamiltonians in the appropriate local coordinates. We then rewrite these local
Hamiltonians in the molecular or laboratory representation, along with the
Dzyaloshinskii-Moriay (DM) and isotropic Heisenberg, biquadratic, and
three-center quartic Hamiltonians. Using our exact, compact forms for the
single-ion spin matrix elements, we evaluate the eigenstate energies
analytically to first order in the microscopic anisotropy interactions,
corresponding to the strong exchange limit, and provide tables of simple
formulas for the energies of the lowest four eigenstate manifolds of
ferromagnetic (FM) and anitiferromagnetic (AFM) tetramers with arbitrary $s_1$.
For AFM tetramers, we illustrate the first-order level-crossing inductions for
$s_1=1/2,1,3/2$, and obtain a preliminary estimate of the microscopic
parameters in a Ni$_4$ from a fit to magnetization data.
  Accurate analytic expressions for the thermodynamics, electron paramagnetic
resonance absorption and inelastic neutron scattering cross-section are given,
allowing for a determination of three of the microscopic anisotropy
interactions from the second excited state manifold of FM tetramers. We also
predict that tetramers with symmetries $S_4$ and $D_{2d}$ should exhibit both
DM interactions and multiferroic states, and illustrate our predictions for
$s_1=1/2, 1$.
