  For the solution of the free Schr\"odinger equation, we obtain the optimal
constants and characterise extremisers for forward and reverse smoothing
estimates which are global in space and time, contain a homogeneous and radial
weight in the space variable, and incorporate a certain angular regularity.
This will follow from a more general result which permits analogous sharp
forward and reverse smoothing estimates and a characterisation of extremisers
for the solution of the free Klein-Gordon and wave equations. The nature of
extremisers is shown to be sensitive to both the dimension and the size of the
smoothing index relative to the dimension. Furthermore, in four spatial
dimensions and certain special values of the smoothing index, we obtain an
exact identity for each of these evolution equations.
