Inferring features about the spacetime from a line element

Suppose $g_{ij}$ is a symmetric tensor describing the geometry of a 4d spacetime. Then I know for example, that if the metric contains "mixed terms" (like $dtdx$) then this spacetime is not static, provided also that the metric is invariant under $t mapsto -t$ (although I’ve read that this is not strictly true). I was wondering if it is possible to deduce other features of the spacetime from the metric, for example would it be possible to deduce the spacetime is isotropic or homogeneous from the line element?