  Fracton order describes novel quantum phases of matter that host
quasiparticles with restricted mobility, and thus lies beyond the existing
paradigm of topological order. In particular, excitations that cannot move
without creating multiple excitations are called fractons. Here we address a
fundamental open question -- can the notion of self-exchange statistics be
naturally defined for fractons, given their complete immobility as isolated
excitations? Surprisingly, we demonstrate how fractons can be exchanged, and
show that their self-statistics is a key part of the characterization of
fracton orders. We derive general constraints satisfied by the fracton
self-statistics in a large class of Abelian fracton orders. Finally, we show
the existence of nontrivial fracton self-statistics in some twisted variants of
the checkerboard model and Haah's code, establishing that these models are in
distinct quantum phases as compared to their untwisted cousins.
