I recently read articles on paleomagnetism. It is common to make the assumption of antipodal symmetry for the distribution of the dispersion of the directions. Two such spherical distributions (directional statistics) with such an antipodal symmetry are the Bingham and the Kent distributions. It will be worth writting the jMEF files for those ones.

Frank.

They are only 6 exponential family distributions that admit variance as a quadratic function (QVF=quadratic variance function) of the parameter. For the multivariate case, it is a bit more complex but well defined and studied:

The $2d+4$ simple quadratic natural exponential families on $R^d$

Frank.