A Symmetry in Covariant Derivatives?
A covariant derivative is the sum of two
parts that are not tensors:
 is all about changes in the potential.

is all about changes in the metric.
Until something is specified about either the potential or the metric, a
covariant derivative could be any continuous combination of the
change in potential and change in the metric.
Does that sound like a symmetry?
Is there a name already for this?