  We investigate properties of non-translation-invariant measures, describing
particle systems on $\bbz$, which are asymptotic to different translation
invariant measures on the left and on the right. Often the structure of the
transition region can only be observed from a point of view which is
random---in particular, configuration dependent. Two such measures will be
called shift equivalent if they differ only by the choice of such a viewpoint.
We introduce certain quantities, called translation sums, which, under some
auxiliary conditions, characterize the equivalence classes. Our prime example
is the asymmetric simple exclusion process, for which the measures in question
describe the microscopic structure of shocks. In this case we compute
explicitly the translation sums and find that shocks generated in different
ways---in particular, via initial conditions in an infinite system or by
boundary conditions in a finite system---are described by shift equivalent
measures. We show also that when the shock in the infinite system is observed
from the location of a second class particle, treating this particle either as
a first class particle or as an empty site leads to shift equivalent shock
measures.
