  We study compactifications on ${\rm AdS}_3\times S^3$ and deformations
thereof. We exploit the triality symmetry of the underlying duality group ${\rm
SO}(4,4)$ of three-dimensional supergravity in order to construct and relate
new consistent truncations. For non-chiral $D=6$, ${\cal N}_{\rm 6d}=(1,1)$
supergravity, we find two different consistent truncations to three-dimensional
supergravity, retaining different subsets of Kaluza-Klein modes, thereby
offering access to different subsectors of the full nonlinear dynamics. As an
application, we construct a two-parameter family of ${\rm AdS}_3\times M^3$
backgrounds on squashed spheres preserving ${\rm U}(1)^2$ isometries. For
generic value of the parameters, these solutions break all supersymmetries, yet
they remain perturbatively stable within a non-vanishing region in parameter
space. They also contain a one-parameter family of ${\cal N}=(0,4)$
supersymmetric ${\rm AdS}_3\times M^3$ backgrounds on squashed spheres with
${\rm U}(2)$ isometries. Using techniques from exceptional field theory, we
determine the full Kaluza-Klein spectrum around these backgrounds.
