  The weakest known criterion for local rotational symmetry (LRS) in spacetimes
of Petrov type D is due to Goode and Wainwright (1986). Here it is shown, using
methods related to the Cartan-Karlhede procedure, to be equivalent to local
spatial rotation invariance of the Riemann tensor and its first derivatives.
Conformally flat spacetimes are similarly studied and it is shown that for
almost all cases the same criterion ensures LRS. Only for conformally flat
accelerated perfect fluids are three curvature derivatives required to ensure
LRS, showing that Ellis's original condition for that case is necessary as well
as sufficient.
