Generalizing the idea of “symmetric distribution” from 1D to multivariate is quite interesting and not trivial.

Here are some possible generalizetions:

Spherical Symmetry

\(X - \theta \sim A (X - \theta)\) For a rotation matrix $A$.

Elliptical Symmetry

Central Symmetry

A.k.a. “reflectively” or “diagonally” or “simply” or “antipodally” symmetric: \(X - \theta \sim \theta - X\)

Angular Symmetry

[1] Serfling, Robert J. “Multivariate Symmetry and Asymmetry.” In Encyclopedia of Statistical Sciences. John Wiley & Sons, Inc., 2004. http://onlinelibrary.wiley.com/doi/10.1002/0471667196.ess5011/abstract.