The structures of space and time associated with various four-dimensional symmetries are constructed and examined. It is interesting that some of these four-dimensional symmetries are compatible with the existence of an aether in the Einstein sense. There are two classes of infinitely many four-dimensional symmetries that cannot be ruled out by previous experiments. We discuss some laser experiments that test these two classes of four-dimensional symmetries. An interesting connection between the nonclassical aether and gravity through the equality of inertial and (...) gravitational mass is explored. (shrink)