  We present a new application of the Regge-Teitelboim method for treating
symmetries which are defined asymptotically. It may be regarded as
complementary to the one in their original 1974 paper. The formulation is based
on replacing an asymptotic plane by the two--sheeted ``hourglass" shaped
surface obtained by joining smoothly an incoming hyperboloid with an outgoing
one. The hyperboloids have a fixed radius, and as one moves the center of the
hourglass along the time axis one covers the whole of spacetime. The motivation
is to study radiation, and the hourglass is well suited to the task because it
is asymptotically null, and thus is able to register the details of the
process. A simple parity condition for the fields on the hyperboloid is given.
It specifies that as much radiation as is coming in as it is going out. With
it, a Hamiltonian formulation of the symmetry of Bondi, van der Burg, Metzner
and Sachs is developed fir both electromagnetism and gravitation. It is
indispensable for the construction to have electric--magnetic duality
asymptotically. For gravitation, a formulation for the linearized theory on the
hourglass has not been explicitly constructed; but enough rudiments of it are
given so that the main results can be established. A definition for angular
momentum wish is conserved (for which the ``magnetic sector'' is essential) is
given. It incorporates an interrelationship between spin and charge. For the
gravitational field, Taub-NUT space appears as the analog of a magnetic pole.
