  We study the asymptotic symmetries of Einstein gravity in flat space. Instead
of Bondi gauge, we work with the recently introduced special double null gauge,
in which $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ are approached along null
directions. We find four new functions worth of asymptotic diffeomorphisms
beyond the familiar supertranslations and superrotations, which are of
relevance in discussions of finite surface charges. Two of these arise from
angle-dependent shifts in the $v$-coordinate near $\mathscr{I}^{+}$. We call
these hypertranslations and sub-leading hypertranslations, with analogous
statements in the $u$-coordinate near $\mathscr{I}^{-}$. There are also two
Diff$(S^2)$ transformations, which we call hyperrotations, that are sub-leading
to the Virasoro superrotations. With power law fall-offs in the null coordinate
and the standard metric on the sphere at leading order, we prove that this is
the exhaustive list of diffeomorphisms whose associated metric parameters can
show up in the (finite) surface charges. We compute the algebra of the
asymptotic Killing vectors under the Barnich-Troessaert bracket, and find a
four-fold infinite generalization of the BMS algebra.
