  We present the generalized Scherk-Schwarz reduction ansatz for the full
supersymmetric exceptional field theory in terms of group valued twist matrices
subject to consistency equations. With this ansatz the field equations
precisely reduce to those of lower-dimensional gauged supergravity parametrized
by an embedding tensor. We explicitly construct a family of twist matrices as
solutions of the consistency equations. They induce gauged supergravities with
gauge groups SO(p,q) and CSO(p,q,r). Geometrically, they describe
compactifications on internal spaces given by spheres and (warped)
hyperboloides $H^{p,q}$, thus extending the applicability of generalized
Scherk-Schwarz reductions beyond homogeneous spaces. Together with the
dictionary that relates exceptional field theory to D=11 and IIB supergravity,
respectively, the construction defines an entire new family of consistent
truncations of the original theories. These include not only compactifications
on spheres of different dimensions (such as AdS$_5\times S^5$), but also
various hyperboloid compactifications giving rise to a higher-dimensional
embedding of supergravities with non-compact and non-semisimple gauge groups.
