  We introduce and develop a theory of fusion and statistical processes of
gapped excitations in Abelian fracton phases. The key idea is to incorporate
lattice translation symmetry via its action on superselection sectors, which
results in a fusion theory endowed with information about the non-trivial
mobility of fractons and sub-dimensional excitations. This results in a
description of statistical processes in terms of local moves determined by the
fusion theory. Our results can be understood as providing a characterization of
translation-invariant fracton phases. We obtain simple descriptions of the
fusion theory in the X-cube and checkerboard fracton models, as well as for
gapped electric and magnetic excitations of some gapless U(1) tensor gauge
theories. An alternate route to the X-cube model fusion theory is provided by
starting with a system of decoupled two-dimensional toric code layers, and
giving a description of the p-string condensation mechanism within our
approach. We discuss examples of statistical processes of fractons and
sub-dimensional excitations in the X-cube and checkerboard models. As an
application of the ideas developed, we prove that the X-cube and semionic
X-cube models realize distinct translation-invariant fracton phases, even when
the translation symmetry is broken corresponding to an arbitrary but finite
enlargement of the crystalline unit cell.
