  In this paper a hidden extra symmetry of conformally invariant Lagrangians
occuring in physics is pointed out. This symmetry is most apparent in a metric
independent, i.e. in a Palatini-like presentation of the variational problem.
In such presentation, the usual conformal weight of fields can be encoded as
local dilatation group gauge charges. The conventional conformal invariance of
Lagrangians is then equivalent to dilatation gauge invariance. The claim of the
paper is, that the most commonly occurring conformally invariant Lagrangians
turning up in physics are not only invariant to local dilatation gauge
transformations, but they are also invariant to any change of the dilatation
gauge connection, meaning an additional algebraic symmetry property. In terms
of dimensional analysis and differential geometry, this additional symmetry
means complete insensitivity of the Lagrangian to the choice of the parallel
transport rule of local measurement units throughout spacetime.
