  The Ricci and energy-momentum tensors have the same algebraic symmetries. In
the Einstein equations they look ``dual'' to each other, in that interchanging
them and inverting the gravitational coupling leaves the equations invariant.
It may then be expected that their differential symmetry Lie algebras would
also be identical. Using cylindrically symmetric static spacetimes it is shown
that they are not identical and neither algebra is a subset of the other.
